CHAPTER. 11 Circles Carnegie Learning, Inc.

Transcription

1 CHAPTER Circles Gears are circular objects used to transmit rotational forces from one mechanical device to another. Often, gears are used to speed up or slow down the rate of rotation of a mechanical object, such as a driveshaft or axle. Interlocking gears of different sizes revolve at different rates, with a larger gear completing a full revolution more slowly than a smaller one. You will use the arc length of a circle to better understand how gears work together..1 Riding a Ferris Wheel Introduction to Circles p Take the Wheel Central Angles, Inscribed Angles, and Intercepted Arcs p Manhole Covers Measuring Angles Inside and Outside of Circles p Color Theory Chords p Solar Eclipses Tangents and Secants p Replacement for a Carpenter s Square Inscribed and Circumscribed Triangles and Quadrilaterals p Gears Arc Length p Playing Darts Sectors and Segments of a Circle p The Coordinate Plane Circles and Polygons on the Coordinate Plane p. 713 Chapter Circles 633

2 Introductory Problem for Chapter That Darn Kitty! A neighbor gave you a plate of cookies as a housewarming present. Before you could eat a single cookie, the cat jumped onto the kitchen counter and knocked the cookie plate onto the floor, shattering it into many pieces. The cookie plate will need to be replaced and returned to the neighbor. Unfortunately, cookie plates come in various sizes and you need to know the exact diameter of the broken plate. It would be impossible to reassemble all of the broken pieces, but one large chunk has remained intact as shown. You think to yourself that there has to be an easy way to determine the diameter of the broken plate. As you sit staring at the large piece of the broken plate, your sister Sarah comes home from school. You update her on the latest crisis and she begins to smile. Sarah tells you not to worry because she learned how to solve for the diameter of the plate in geometry class today. She gets a piece of paper, a compass, a straightedge, a ruler, and a marker out of her backpack and says, Watch this! What does Sarah do? Describe how she can determine the diameter of the plate with the broken piece. Then, show your work on the broken plate shown. 634 Chapter Circles

3 .1 Riding a Ferris Wheel Introduction to Circles OBJECTIVES In this lesson you will: Identify parts of a circle. Draw parts of a circle. KEY TERMS center of a circle chord secant of a circle tangent of a circle point of tangency central angle inscribed angle arc major arc minor arc semicircle The first Ferris wheel was built in 1893 for the Chicago World s Fair to rival the Eiffel Tower, which was built for the Paris World s Fair. Today, the Sky Dream Fukuoka Ferris wheel in Japan is the world s largest Ferris wheel. Lesson.1 Riding a Ferris Wheel 635

4 PROBLEM 1 Going Around and Around A Ferris wheel is in the shape of a circle. Recall, a circle is the set of all points in a plane that are equidistant from a given point, which is called the center of the circle. The distance from a point on the circle to the center is the radius of the circle. A circle is named by its center. For instance, the circle seen in the Ferris wheel is circle P. 1. Use the circle to answer the questions. a. Name the circle. A Recall that a circle is a locus of points on a plane equidistant from a given point. Because the points in a locus are infinite in number, they merge together and appear to resemble a curve. If you used the world s most powerful magnifying glass and focused it on any portion of the curve, you would see millions and millions of microscopic points. Only those points are considered to be the actual circle. O D E B C F 636 Chapter Circles Points associated with a circle appear in one of three possible regions: points are located in the interior of a circle, points are located on the circle, or points are located in the exterior of a circle.

5 b. For each of the points associated with the circle shown, identify which points are located in the interior of the circle, on the circle, or in the exterior of the circle. Which point appears to be the center point of the circle? c. Use a straightedge to draw OB, the radius of circle O. Where are the endpoints located with respect to the circle? Take Note Radii is the plural of radius. d. How many radii does a circle have? Explain your reasoning. e. Use a straightedge to draw AC. Then use a straightedge to draw BD. How are the line segments different? How are they the same? Both line segments are chords of the circle because the endpoints are on the circle. Segment AC is called a diameter of the circle. f. Why is BD not considered a diameter? Lesson.1 Riding a Ferris Wheel 637

6 g. How does the length of the diameter of a circle relate to the length of the radius? h. Are all radii of the same circle, or of congruent circles (always, sometimes, or never), congruent? Explain. A secant of a circle is a line that intersects a circle at exactly two points. 2. Draw a secant using the circle shown. Z 3. Explain the difference between a chord and a secant. 4. Why is the diameter of a circle considered the longest chord in a circle? A tangent of a circle is a line that intersects a circle at exactly one point. The point of intersection is called the point of tangency. 638 Chapter Circles

7 5. Draw a tangent using circle Z shown. Z 6. Choose another point on the circle. How many lines can you draw through this point tangent to the circle? 7. Explain the difference between a secant and a tangent. 8. Check the appropriate term(s) associated with each characteristic in the table shown. Characteristic Chord Secant Diameter Radius Tangent A line A line segment A line segment having both endpoints on the circle A line segment having one endpoint on the circle A line segment passing through the center of the circle A line intersecting a circle at exactly two points A line intersecting a circle at exactly one point Lesson.1 Riding a Ferris Wheel 639

8 PROBLEM 2 Sitting on the Wheel A central angle is an angle of a circle whose vertex is the center of the circle. An inscribed angle is an angle of a circle whose vertex is on the circle. 1. Four friends are riding a Ferris wheel in the positions shown. Dru Marcus O Wesley Kelli a. Draw a central angle where Dru and Marcus are located on the sides of the angle. b. Draw an inscribed angle where Kelli is the vertex and Dru and Marcus are located on the sides of the angle. c. Draw an inscribed angle where Wesley is the vertex and Dru and Marcus are located on the sides of the angle. d. Compare and contrast these three angles. An arc of a circle is any unbroken part of the circumference of a circle. An arc is named using its two endpoints. The symbol used to describe arc AB is AB. A major arc of a circle is the largest arc formed by a secant and a circle. It goes more than half way around a circle. A minor arc of a circle is the smallest arc formed by a secant and a circle. It goes less than half way around a circle. A semicircle is exactly half of a circle. 640 Chapter Circles

9 2. A copy of the Ferris wheel from Question 1 is shown. O a. Label the location of each person with the first letter of his or her name. b. Use your pencil to trace DM. Describe what you traced. c. Use your pencil to trace MD. Describe what you traced. d. What is the difference between DM and MD? e. Draw a diameter on the circle shown so that point D is an endpoint. Label the second endpoint as point Z. The diameter divided the circle into two semicircles. f. Use your pencil to trace one semicircle. Describe what you traced. g. Use your pencil to trace a different semicircle. Describe what you traced. To avoid confusion, three points are used to name semicircles and major arcs. The first point is an endpoint of the arc, the second point is any point at which the arc passes through and the third point is the other endpoint of the arc. h. Name each semicircle. i. Name all minor arcs. j. Name all major arcs. Lesson.1 Riding a Ferris Wheel 641

10 PROBLEM 3 Name the Parts Use the diagram shown to answer Questions 1 through Name a diameter. 2. Name a radius. E C O I 3. Name a central angle. L R 4. Name an inscribed angle. 5. Name a minor arc. 6. Name a major arc. 7. Name a semicircle. Be prepared to share your solutions and methods. 642 Chapter Circles

11 .2 Take the Wheel Central Angles, Inscribed Angles, and Intercepted Arcs OBJECTIVES In this lesson you will: Determine the measures of arcs. Use the Arc Addition Postulate. Determine the measures of central angles and inscribed angles. Prove the Inscribed Angle Theorem. Prove the Parallel Lines-Congruent Arcs Theorem. KEY TERMS degree measure (of an arc) adjacent arcs Arc Addition Postulate intercepted arc Inscribed Angle Theorem Parallel Lines-Congruent Arcs Theorem Before airbags were installed in car steering wheels, the recommended position for holding the steering wheel was the 10-2 position. Now, one of the recommended positions is the 9-3 position to account for the airbags. The numbers 10, 2, 9, and 3 refer to the numbers on a clock. So the 10-2 position means that one hand is at 10 o clock and the other hand is at 2 o clock. Lesson.2 Take the Wheel 643

12 PROBLEM 1 Keep Both Hands on the Wheel The circles shown represent steering wheels, and the points on the circles represent the positions of a person s hands. A B C D O P Recall that the degree measure of a circle is 360. Each minor arc of a circle is associated with and determined by a specific central angle. The degree measure of a minor arc is the same as the degree measure of its central angle. For example, if the measure of central angle PRQ is 30 then the degree measure of its minor arc PQ is equal to 30. Using symbols, this can be expressed as follows: If PRQ is a central angle and m PRQ 30, then m PQ For each circle, use the given points to draw a central angle. The hand position on the left is 10-2 and the hand position on the right is -1. What are the names of the central angles? 2. Without using a protractor, determine the central angle measures. Explain your reasoning. 3. How do the measures of these angles compare? 4. Why do you think the hand position represented by the circle on the left was recommended and the hand position represented on the right is not recommended? 644 Chapter Circles

13 5. Describe the measures of the minor arcs you named in Question 3 using symbols. 6. Plot and label point Z on each circle so that it does not lie between the endpoints of the minor arcs you identified in Question 5. Determine the measures of the major arcs that have the same endpoints as the minor arcs in Question 5. Explain your reasoning. 7. What is the measure of a semicircle? Explain your reasoning. 8. If the measures of two central angles of the same circle (or congruent circles) are equal, are their corresponding minor arcs congruent? Explain. 9. If the measures of two minor arcs of the same circle (or congruent circles) are equal, are their corresponding central angles congruent? Explain. Adjacent arcs are two arcs of the same circle sharing a common endpoint. 10. Draw and label two adjacent arcs on circle O shown. O Lesson.2 Take the Wheel 645

14 The Arc Addition Postulate states: The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.. Apply the Arc Addition Postulate to the adjacent arcs you created for Question Name two other postulates you have studied that are similar to the Arc Addition Postulate. An intercepted arc is an arc associated with and determined by angles of the circle. An intercepted arc is a portion of the circumference of the circle located on the interior of the angle whose endpoints lie on the sides of an angle. O 13. Draw inscribed PSR on circle O. 14. Name the intercepted arc associated with PSR. 646 Chapter Circles

15 15. Consider the central angle that is shown. Use a straightedge to draw an inscribed angle that contains points A and B on its sides. Name the vertex of your angle point P. A O B a. What do the angles have in common? b. Use your protractor to measure the central angle and the inscribed angle. How is the measure of the inscribed angle related to the measure of the central angle and the measure of AB? c. Use a straightedge to draw a different inscribed angle that contains points A and B on its sides. Name its vertex point Q. Measure the inscribed angle. How is the measure of the inscribed angle related to the measure of the central angle and the measure of AB? d. Use a straightedge to draw one more inscribed angle that contains points A and B on its sides. Name its vertex point R. Measure the inscribed angle. How is the measure of the inscribed angle related to the measure of the central angle and the measure of AB? 16. What can you conclude about inscribed angles that have the same intercepted arc? Lesson.2 Take the Wheel 647

16 17. Create an Inscribed Angle Conjecture about the measure of an inscribed angle and the measure of its intercepted arc. 18. Inscribed angles formed by two chords can be drawn three different ways with respect to the center of the circle. a. Case 1: Use circle O shown to draw and label inscribed MPT such that the center point lies on one side of the inscribed angle. O b. Case 2: Use circle O shown to draw and label inscribed MPT such that the center point lies on the interior of the inscribed angle. O c. Case 3: Use circle O shown to draw and label inscribed MPT such that the center point lies on the exterior of the inscribed angle. O 648 Chapter Circles

17 19. To prove your Inscribed Angle Conjecture, you must prove each case in Question 18, parts (a) through (c). a. Use the diagram provided to complete the proof for Case 1. M O P T Given: MPT is inscribed in circle O. m MPT x Point O lies on diameter PM. Prove: m MPT 1 2 m MT Statements 1. MPT is inscribed in circle O. m MPT x Point O lies on diameter PM. 2. Connect points O and T to form radius OT. 1. Given 2. Construction Reasons Lesson.2 Take the Wheel 649

18 b. Use the diagram provided to complete the proof for Case 2. M R T O P Given: MPT is inscribed in circle O. Point O is in the interior of MPT. m MPO x m TPO y Prove: m MPT 1 2 m MT Statements 1. MPT is inscribed in circle O. Point O is in the interior of MPT. m MPO x m TPO y 2. Construct diameter PR. Connect points O and T to form radius OT. Connect points O and M to form radius OM. 1. Given Reasons 2. Construction 650 Chapter Circles

19 c. Use the diagram provided to complete the proof for Case 3. Write a paragraph proof or create a two-column proof. Hint: You will need to construct a diameter through point P and construct radii OM and OT. Given: MPT is inscribed in circle O. Point O is in the exterior of MPT. Prove: m MPT 1 2 m MT R O P M T Statements Reasons Lesson.2 Take the Wheel 651

20 Congratulations! You have just proven the Inscribed Angle Conjecture. It is now known as the Inscribed Angle Theorem. You can now use this theorem as a valid reason in proofs. The Inscribed Angle Theorem states: The measure of an inscribed angle is one-half the measure of its intercepted arc. PROBLEM 2 Parallel Lines Intersecting a Circle Allisa was excited because she invented her own conjecture in geometry class! The focus of the lesson was the relationship between an inscribed angle and the measure of its intercepted arc. Her teacher told the class to discover something they didn t already know using what they learned in class. Allisa was busy drawing different kinds of diagrams and she decided to draw a circle with two parallel lines as shown. A O R P L Allisa quickly wrote the following conjecture: Parallel lines intercept congruent arcs on a circle. She named her conjecture the Parallel Lines-Congruent Arcs Conjecture. 652 Chapter Circles

21 1. Create a two-column proof of the Parallel Lines-Congruent Arcs Conjecture. Given: Prove: Statements Reasons Congratulations! You have just proven the Parallel Lines-Congruent Arcs Conjecture. It is now known as the Parallel Lines-Congruent Arcs Theorem. You can now use this theorem as a valid reason in proofs. PROBLEM 3 Determine the Measure MP is a diameter of circle O. 1. If m MT 124, determine m TPW. Explain your reasoning. M K O T S P W Lesson.2 Take the Wheel 653

22 Use the diagram to answer Questions 2 through 4. F J K G O 2. Are radii OJ, OK, OF, and OG all congruent? Explain your reasoning. 3. Is m FG greater than m JK? Explain your reasoning. 4. If m FOG 57, determine m JK and m FG. Explain your reasoning. 5. DeJaun told Thomas there was not enough information to determine whether circle A was congruent to circle B. He said they would have to know the length of a radius in each circle to determine whether the circles were congruent. Thomas explained to DeJaun why he was incorrect. What did Thomas say to DeJaun? Be prepared to share your solutions and methods. A B 654 Chapter Circles

23 .3 OBJECTIVES In this lesson you will: Manhole Covers Measuring Angles Inside and Outside of Circles Determine measures of angles formed by two chords. Determine measures of angles formed by two secants. Determine measures of angles formed by a tangent and a secant. Determine measures of angles formed by two tangents. Prove the Interior Angles of a Circle Theorem. Prove the Exterior Angles of a Circle Theorem. KEY TERMS Interior Angles of a Circle Theorem Exterior Angles of a Circle Theorem Manhole covers are heavy removable plates that are used to cover maintenance holes in the ground. Most manhole covers are circular and can be found all over the world. The tops of these covers can be plain or have beautiful designs cast into their tops. Lesson.3 Manhole Covers 655

24 PROBLEM 1 Inside the Circle Circle O shows a simple manhole cover design. A E O B D m BD 70 m AC 0 C 1. Consider BED. How is this angle different from the angles that you have seen so far in this chapter? How is this angle the same? 2. Can you determine the measure of BED with the information you have so far? If so, how? Explain your reasoning. 3. Draw chord CD. Use the information given in the figure to name the measures of any angles that you do know. Explain your reasoning. 4. How does BED relate to CED? 5. Write a statement showing the relationship between m BED, m EDC, and m ECD. 6. What is the measure of BED? 7. Consider circle P shown. Draw chord XY on the figure shown. V P W Z X Y a. Write a statement for m WXY in terms of m WY. b. Write a statement m VYX for in terms of m VX. 656 Chapter Circles

25 c. Write a statement for m WZY in terms of m WY and m VX. The measure of an angle formed by two intersecting chords is half of the sum of the measures of the arcs intercepted by the angle and its vertical angle. d. Consider WZY again. What is the arc that is intercepted by WZY? e. Name the angle that is vertical to WZY. Then name the arc that is intercepted by the angle vertical to WZY. 8. Consider the figure. By the statement you wrote previously, you can state that CEB 1 2 (m CB m AD ). Write similar statements for m AED, AEC, and DEB. D A C O E B The Interior Angles of a Circle Theorem states: If an angle is formed by two intersecting chords or secants of a circle such that the vertex of the angle is in the interior of the circle, then the measure of the angle is half of the sum of the measures of the arcs intercepted by the angle and its vertical angle. Lesson.3 Manhole Covers 657

26 9. Create a two-column proof of the Interior E Angles of a Circle Theorem. Given: Chords EK and GH intersect at point F in circle O. Prove: m KFH 1 2 (m HK m EG ) O F G K H Statements 1. Chords EK and GH intersect at point F in circle O. 2. Connect points E and H to form chord EH. 1. Given 2. Construction Reasons PROBLEM 2 Outside the Circle Circle T shows another simple manhole cover design. K L T N M m KM 80 m LN Consider KL and MN. Use a straightedge to draw secants that coincide with each segment. Where do the secants intersect? Label this point as point P on the figure. 2. Draw chord KN. Can you determine the measure of KPM with the information you have so far? If so, how? Explain your reasoning. 658 Chapter Circles

27 3. Use the information given in the figure for Problem 2 to name the measures of any angles that you do know. Explain how you determined your answers. 4. How does KPN relate to KPN? 5. Write a statement showing the relationship between m KPN, m NKP, and m KNM. 6. What is the measure of KPN? 7. Describe the measure of KPM in terms of the measures of both arcs intercepted by KPM. If an angle is formed by two intersecting secants so that the angle is outside the circle, then the measure of this angle is half of the difference of the arc measures that are intercepted by the angle. 8. Consider the figure. Use the previous statement to write a statement for m CAD in terms of the arc measures that are intercepted by CAD. A B C E O D Lesson.3 Manhole Covers 659

28 9. In circle P, the line through point T is tangent to the circle and is perpendicular to UT. U V T W a. What are the measures of UXT and UYT? X P Y b. What are the measures of UTV and UTW? c. Do you think that there is a relationship between m UXT and m UTV? Do you think that there is a relationship between m UYT and m UTW? If so, what is the relationship? 10. Line TS is tangent to circle Q. R Q S a. Use a straightedge to draw the central angle that is associated with RT. Then use your protractor to measure RQT and RTS. b. How do the measures of the angles compare?. Create an Exterior Angles of a Circle Conjecture about the angle measure that is formed by a tangent and a chord of a circle and the arc measure intercepted by the chord. T 660 Chapter Circles

29 12. Suppose that a tangent and a secant to a circle intersect, as shown. First, draw a chord that connects point Q and point T on the circle. Then, use an argument similar to the one in Question 8 to show that m QST 1 2 (m QT m RT ). Q O T R S U 13. Suppose that two tangents to a circle intersect, as shown. First, draw a chord that connects point B and point D on the circle. Then use an argument similar to the one in Question 8 to show that m BCD 1 2 (m BGD m BD ). A B G O C D E Lesson.3 Manhole Covers 661

30 PROBLEM 3 Proving Conjectures 1. An angle with a vertex located in the exterior of a circle can be formed by a secant and a tangent, two secants, or two tangents. a. Case 1: Use circle O shown to draw and label an exterior angle formed by a secant and a tangent. O b. Case 2: Use circle O shown to draw and label an exterior angle formed by two secants. O c. Case 3: Use circle O shown to draw and label an exterior angle formed by two tangents. O To prove the Exterior Angles of a Circle Conjecture previously stated, you must prove each of the three cases. 662 Chapter Circles

31 2. Complete the proof for each case. a. Use the diagram provided to complete the proof for Case 1. E O R T X A Given: Secant EX and tangent TX intersect at point X. Prove: m EXT 1 2 (m ET m RT ) Statements 1. Secant EX and tangent TX intersect at point X. 2. Connect points E and T to form chord ET. Connect points R and T to form chord RT. 1. Given 2. Construction Reasons Lesson.3 Manhole Covers 663

32 b. Use the diagram provided to complete the proof for Case 2. A X E O T R Given: Secants EX and RX intersect at point X. Prove: m EXR 1 2 (m ER m AT ) Statements 1. Secants EX and TX intersect at point X. 2. Connect points A and R to form chord AR. 1. Given 2. Construction Reasons 664 Chapter Circles

33 c. Use the diagram provided to complete the proof for Case 3. E X O T R A Given: Tangents EX and AX intersect at point X. Prove: m EXT 1 2 (m ERT m ET ) Statements Reasons 1. Tangents EX and TX intersect at point X. 1. Given Congratulations! You have just proven the Exterior Angles of a Circle Conjecture. It is now known as the Exterior Angles of a Circle Theorem. You can use this theorem as a valid reason in proofs. The Exterior Angles of a Circle Theorem states: If an angle is formed by two intersecting secants, two intersecting tangents, or an intersecting tangent and secant of a circle such that the vertex of the angle is in the exterior of the circle, then the measure of the angle is half of the difference of the measures of the arc(s) intercepted by the angle. Lesson.3 Manhole Covers 665

34 PROBLEM 4 Determine the Measures 1. Use the diagrams shown to determine the measures of each. a. Determine m RT. m FG 86 m HP 21 F 86 G R T H 21 P W b. Using the given information, what additional information can you determine about the diagram? Chapter Circles

35 c. Determine m CD. m AB 88 m AED 80 C 88 B E 80 D A d. Explain how knowing m ERT can help you determine m EXT. E X R T Be prepared to share your solutions and methods. Lesson.3 Manhole Covers 667

36 668 Chapter Circles

37 .4 Color Theory Chords OBJECTIVES In this lesson you will: Determine the relationships between a chord and a diameter of a circle. Determine the relationships between congruent chords and their minor arcs. Prove the Diameter-Chord Theorem. Prove the Equidistant Chord Theorem. Prove the Equidistant Chord Converse Theorem. Prove the Congruent Chord-Congruent Arc Theorem. Prove the Congruent Chord-Congruent Arc Converse Theorem. Prove the Segment-Chord Theorem. KEY TERMS Diameter-Chord Theorem Equidistant Chord Theorem Equidistant Chord Converse Theorem Congruent Chord-Congruent Arc Theorem Congruent Chord-Congruent Arc Converse Theorem segments of a chord Segment-Chord Theorem 1 Color theory is a set of rules that is used to create color combinations. A color wheel is a visual representation of color theory. There are many kinds of color wheels; consider the diagram as an RYB (red-yellow-blue) color wheel. R: Red Y: Yellow B: Blue P: Purple O: Orange G: Green The color wheel is made of three different kinds of colors: primary, secondary, and tertiary. Primary colors (red, blue, and yellow) are the colors you start with. Secondary colors (orange, green, and purple) are created by mixing two primary colors. Tertiary colors (red-orange, yellow-orange, yellow-green, blue-green, bluepurple, red-purple) are created by mixing a primary color with a secondary color. Lesson.4 Color Theory 669

38 PROBLEM 1 Mixing Primary Colors 1. The locations of the primary colors on the color wheel are points Y (yellow), R (red), and B (blue) as shown on the circle. Y B R a. Use a straightedge to draw a chord that has endpoints that are primary colors. What color is created if you mix these two colors? b. Use your compass and straightedge to draw the perpendicular bisector of the chord. c. What do you notice about your perpendicular bisector? 2. Draw a chord AB that does not pass through the center of circle T. Then, use the following steps to draw a diameter that is perpendicular to chord AB. T a. Place your compass point on the center of circle T. Draw an arc that intersects the chord at two points. Name these point C and point D. Now open your compass wider than half the distance between CD. Place the point of the compass on point C and draw an arc toward the center of the circle. Place the point of the compass on point D and draw an arc toward the center of the circle. Use your straightedge to draw the diameter that passes through the intersection of the arcs. Label the point where the diameter intersects the chord as point P. Label the point where the diameter intersects AB as point Q. b. How does the length of AP compare to the length of PB? What does this tell you about the diameter? 670 Chapter Circles

39 c. How does the measure of AQ compare to the measure of BQ? Explain your reasoning. 3. Write a Diameter-Chord Conjecture about the effect a circle s diameter that is perpendicular to a chord has on that chord and its corresponding arc. Let s prove the conjecture. 4. Create a two-column proof of the Diameter-Chord Conjecture. Given: MI is a diameter of circle O. MI DA Prove: MI bisects DA. MI bisects DA. D O E I A M 1. Statements MI is a diameter of circle O. MI DA 2. Connect points O and D to form chord OD. Connect points O and A to form chord OA. 1. Given 2. Construction Reasons Lesson.4 Color Theory 671

40 Congratulations! You have just proven the Diameter-Chord Conjecture. It is now known as the Diameter-Chord Theorem. You can now use this theorem as a valid reason in proofs. The Diameter-Chord Theorem states: If a circle s diameter is perpendicular to a chord, then the diameter bisects the chord and bisects the arc determined by the chord. 5. What does TP represent in the relationship between point T and chord AB in Question 2? 6. Use a straightedge to draw two congruent chords on circle T that are not parallel to each other. The chords should not be diameters. Label the chords AB and CD. T a. For each chord, use your compass and straightedge to draw a line segment that represents the distance from the center of the circle to the chord. Then use your compass to compare the lengths of these segments. b. What do you notice? 7. Write an Equidistant Chord Conjecture about congruent chords and their distance from the center of the circle. Let s prove the conjecture. 672 Chapter Circles

41 8. Create a two-column proof of the Equidistant Chord Conjecture. H R C E O I D Given: CH DR OE CH OI DR Prove: CH and DR are equidistant from the center point. Hint: You need to get OE OI. Statements CH DR OE CH OI DR Connect points O and H, O and C, O and D, O and R to form radii OH, OC, OD, and OR, respectively. 1. Given 2. Construction Reasons Congratulations! You have just proven the Equidistant Chord Conjecture. It is now known as the Equidistant Chord Theorem. You can now use this theorem as a valid reason in proofs. The Equidistant Chord Theorem states: If two chords of the same circle or congruent circles are congruent, then they are equidistant from the center of the circle. Lesson.4 Color Theory 673

42 Consider the converse of this theorem. The Equidistant Chord Converse Theorem states: If two chords of the same circle or congruent circles are equidistant from the center of the circle, then the chords are congruent. 9. Create a two-column proof of the Equidistant Chord Converse Theorem. H R C E O I D Given: OE OI ( CH and DR are equidistant from the center point.) OE CH OI DR Prove: CH DR Statements 1. OE OI OE CH OI DR 2. Connect points O and H, O and C, O and D, O and R to form radii OH, OC, OD, and OR, respectively. 1. Given 2. Construction Reasons 674 Chapter Circles

43 10. Write the Equidistant Chord Theorem and the Equidistant Chord Converse Theorem as a biconditional statement. PROBLEM 2 Mixing Primary and Secondary Colors The locations of the primary colors on the color wheel are shown on circle T. The primary colors are points R (red), Y (yellow), and B (blue), and the locations of the secondary colors are points O (orange), G (green), and P (purple). Y G O T B R P 1. Use a straightedge to draw two congruent chords. Make sure the chords are not diameters, and so that one endpoint is a primary color and the other endpoint is a secondary color. You can use a compass to verify that the chords are the same length. Write the names of your chords. Identify the colors that would be created if you mixed the colors of the endpoints of each chord. 2. From each endpoint of each chord, use your straightedge to draw a radius. Name the central angle formed by each pair of radii. Use a protractor to find the measures of these central angles. What do you notice? 3. What does Question 2 tell you about the minor arcs formed by the chords? Explain your reasoning. 4. Write a Congruent Chord-Congruent Arc Conjecture about two congruent chords of a circle and the measures of their corresponding arcs. Lesson.4 Color Theory 675

44 5. Create a two-column proof of the Congruent Chord-Congruent Arc Theorem. H R C O D Given: CH DR Prove: CH DR Statements 1. CH DR 2. Connect points O and H, O and C, O and D, O and R to form radii OH, OC, OD, and OR, respectively. 1. Given 2. Construction Reasons Congratulations! You have just proven the Congruent Chord-Congruent Arc Conjecture. It is now known as the Congruent Chord-Congruent Arc Theorem. You can now use this theorem as a valid reason in proofs. The Congruent Chord-Congruent Arc Theorem states: If two chords of the same circle or congruent circles are congruent, then their corresponding arcs are congruent. Consider the converse of this theorem. The Congruent Chord-Congruent Arc Converse Theorem states: If two arcs of the same circle or congruent circles are congruent, then their corresponding chords are congruent. 676 Chapter Circles

45 6. Create a two-column proof of the Congruent Chord-Congruent Arc Converse Theorem. H R C O D Given: CH DR Prove: CH DR Statements 1. CH DR 2. Connect points O and H, O and C, O and D, and O and R to form radii OH, OC, OD, and OR, respectively. 1. Given 2. Construction Reasons 7. Write the Congruent Chord-Congruent Arc Theorem and the Congruent Chord-Congruent Arc Converse Theorem as a biconditional statement. Lesson.4 Color Theory 677

46 PROBLEM 3 Segments Segments of a chord are the segments formed on a chord when two chords of a circle intersect. Use the diagram shown to answer Questions 1 through Name the segments of chord HD. C E H O R D 2. Name the segments of chord RC. 3. Use a ruler to measure the length of each segment of chords HD and RC. 4. What do you notice about the product of the lengths of the segments of chord HD and the product of the lengths of the segments of chord RC? 5. Write a Segment-Chord Conjecture about the products of the lengths of the segments of two chords intersecting in a circle. 678 Chapter Circles

47 6. Create a two-column proof of the Segment-Chord Conjecture. Given: Chords HD and RC intersect at point E in circle O. Prove: EH ED ER EC Hint: Connect points C and D, and points H and R. E Show the triangles are similar. C D H O R Statements Reasons Congratulations! You have just proven the Segment-Chord Conjecture. It is now known as the Segment-Chord Theorem. You can use this theorem as a valid reason in proofs. The Segment-Chord Theorem states: If two chords in a circle intersect, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the second chord. Be prepared to share your solutions and methods. Lesson.4 Color Theory 679

48 680 Chapter Circles

49 .5 OBJECTIVES In this lesson you will: Solar Eclipses Tangents and Secants Determine the relationship between a tangent line and a radius. Determine the relationship between congruent tangent segments. Prove the Tangent Segment Theorem. Prove the Secant Segment Theorem. Prove the Secant Tangent Theorem. KEY TERMS tangent segment Tangent Segment Theorem secant segment external secant segment Secant Segment Theorem Secant Tangent Theorem Total solar eclipses occur when the moon passes between Earth and the sun. The position of the moon creates a shadow on the surface of Earth. A pair of tangent lines forms the boundaries of the umbra, the lighter part of the shadow. Another pair of tangent lines forms the boundaries of the penumbra, the darker part of the shadow. Lesson.5 Solar Eclipses 681

50 PROBLEM 1 Blocking Out the Sun Consider the tangents shown. Sun Moon Earth 1. For the sun and moon, use a straightedge to draw a radius from the center of the circle to one of the tangents. 2. Use your protractor to measure the angle between each radius and tangent. What do you notice? 3. How is the distance from a line to a point not on the line determined? 682 Chapter Circles

51 4. Use a compass and straightedge to determine the distance from AB to point P. P A B 5. Draw a circle and label the center. Then draw a line tangent to the circle on the left or right side of the circle and label the point of tangency. Use a compass and straightedge to determine the distance from the center point to the point of tangency. Lesson.5 Solar Eclipses 683

52 6. Explain why the radius of a circle drawn to the point of tangency is always perpendicular to the tangent line. 7. Can more than one radius of a circle have a perpendicular relationship with a line drawn tangent to the circle? Explain. 8. What can you conclude about a tangent to a circle and a radius drawn to the point of tangency? 9. What do you notice about the tangents that form the umbra in the first diagram? A tangent segment is a segment formed from an exterior point of the circle to the point of tangency. 10. Choose a point exterior to the circle shown. Label this point P. O Use a straightedge to draw a line that passes through point P such that it is tangent to circle O. Draw a different line that passes through point P such that it is also tangent to circle O. Label the points of tangency Q and R. Draw radius OQ and radius OR. Draw a line segment between P and O.. Write a Tangent Segment Conjecture about the tangent segments drawn from the same point on the exterior of a circle. 684 Chapter Circles

53 Let s prove the conjecture. 12. Create a two-column proof of the Tangent Segment Conjecture. T A O N Given: AT is tangent to circle O at point T. AN is tangent to circle O at point N. Prove: AT AN Statements 1. AT is tangent to circle O at point T. AN is tangent to circle O at point N. Reasons Congratulations! You have just proven the Tangent Segment Conjecture. It is now known as the Tangent Segment Theorem. You can use this theorem as a valid reason in proofs. Lesson.5 Solar Eclipses 685

54 The Tangent Segment Theorem states: If two tangent segments are drawn from the same point on the exterior of a circle, then the tangent segments are congruent. 13. In the figure, KP and KS are tangent to circle W and m PKS 46. Calculate m KPS. Explain your reasoning. P W K S 14. In the figure, PS is tangent to circle M and m SMO 9. Calculate m MPS. Explain your reasoning. P S M O 686 Chapter Circles

55 PROBLEM 2 More Segments You have studied angles located in the exterior of a circle. If the angle was formed by two secant lines, then each secant line contains a secant segment and an external secant segment. A secant segment is formed when two secants intersect in the exterior of a circle. A secant segment begins at the point at which the two secants intersect, continues into the circle, and ends at the point at which the secant exits the circle. An external secant segment is the portion of each secant segment that lies on the outside of the circle. It begins at the point at which the two secants intersect and ends at the point where the secant enters the circle. 1. Consider the diagram shown. S O E A C N a. Name two secant segments. b. Name two external secant segments. c. What do you notice about the product of the lengths of a secant segment and its external secant segment and the product of the lengths of the second secant segment and its external secant segment? Lesson.5 Solar Eclipses 687

56 2. Write a Secant Segment Conjecture about the products of the lengths of a secant segment and its external secant segment of two secants intersecting outside a circle. 3. Create a two-column proof of the Secant Segment Conjecture. Given: Secants CS and CN intersect at point C in the exterior of circle O. Prove: CS CE CN CA Hint: Connect points A and S, and points E and N. O Statements Reasons 688 Chapter Circles Congratulations! You have just proven the Secant Segment Conjecture. It is now known as the Secant Segment Theorem. You can use this theorem as a valid reason in proofs. The Secant Segment Theorem states: If two secants intersect in the exterior of a circle, then the product of the lengths of the secant segment and its external secant segment is equal to the product of the lengths of the second secant segment and its external secant segment.

57 4. Consider an angle formed in the exterior of a circle by a secant and a tangent shown in the diagram. T 3 cm A G O 4 cm N 1.5 cm a. Name a tangent segment. b. Name a secant segment and an external secant segment. c. What do you notice about the product of the lengths of a secant segment and its external secant segment and the square of the length of the tangent segment? 5. Write a Secant Tangent Conjecture about the product of the lengths of a secant segment and its external secant segment and the square of the length of the tangent segment. Lesson.5 Solar Eclipses 689

58 6. Create a two-column proof of the Secant Tangent Conjecture. Given: Tangent AT and secant AG intersect at point A in the exterior of circle O. Prove: (AT) 2 AG AN Hint: Connect points N and T, and points G and T. O Statements Reasons Congratulations! You have just proven the Secant Tangent Conjecture. It is now known as the Secant Tangent Theorem. You can use this theorem as a valid reason in proofs. The Secant Tangent Theorem states: If a tangent and a secant intersect in the exterior of a circle, then the product of the lengths of the secant segment and its external secant segment is equal to the square of the length of the tangent segment. Be prepared to share your solutions and methods. 690 Chapter Circles

59 .6 Replacement for a Carpenter s Square Inscribed and Circumscribed Triangles and Quadrilaterals OBJECTIVES In this lesson you will: Determine a property of a triangle inscribed in a circle. Determine how to circumscribe a triangle about a circle. Determine a property of a quadrilateral inscribed in a circle. Determine how to circumscribe a quadrilateral about a circle. Prove the Inscribed Right Triangle- Diameter Theorem. Prove the Inscribed Right Triangle- Diameter Converse Theorem. Prove the Inscribed Quadrilateral- Opposite Angles Theorem. KEY TERMS inscribed polygon Inscribed Right Triangle-Diameter Theorem Inscribed Right Triangle-Diameter Converse Theorem circumscribed polygon Inscribed Quadrilateral-Opposite Angles Theorem A carpenter s square is a tool that is used to create right angles. These squares are usually made of a strong material like metal so that the right angle is not easily bent or broken. Lesson.6 Replacement for a Carpenter s Square 691

60 PROBLEM 1 In Need of a New Tool A carpenter is working on building a children s playhouse. She accidentally drops her carpenter s square and the right angle gets bent. She still needs to cut out a piece of plywood that is in the shape of a right triangle. So the carpenter gets out her compass and straightedge to get the job done. 1. Use the steps to recreate how the carpenter created the right triangle. a. The hypotenuse of the triangle needs to be 6 centimeters. Use your ruler and open your compass to 3 centimeters. In the space provided, draw a circle with a diameter of 6 centimeters. Use the given point as the center. b. Use your straightedge to draw a diameter on the circle. c. One of the legs of the triangle is to be 4 centimeters long. Open your compass to 4 centimeters. Place the point of your compass on one of the endpoints of the diameter and draw an arc that passes through the circle. d. Use your straightedge to draw segments from the endpoints of the diameter to the intersection of the circle and the arc. e. Use your protractor to verify that this triangle is a right triangle. An inscribed polygon is a polygon drawn inside a circle such that each vertex of the polygon touches the circle. 692 Chapter Circles

61 2. Consider ABC that is inscribed in circle P. B a. What do you know about AC? A P D C b. What do you know about m ADC? Explain your reasoning. c. What does this tell you about m ABC? Explain your reasoning. d. What kind of triangle is ABC? How do you know? 3. Write an Inscribed Right Triangle-Diameter Conjecture about the kind of triangle inscribed in a circle when one side of the triangle is a diameter. 4. Write the converse of the conjecture you wrote in Question 3. Do you think this statement is also true? Lesson.6 Replacement for a Carpenter s Square 693

62 5. Use your compass to draw a circle. Then use your protractor and straightedge to draw an inscribed angle in the circle so that the angle is a right angle. Finally, use your straightedge to complete the triangle. a. Is one of the sides a diameter? Which side? b. Consider the intercepted arc of your right angle. What is its measure? Why does this tell you that one of the sides of the triangle must be a diameter? Explain your reasoning. Let s prove the conjectures! 6. Create a two-column proof of the Inscribed Right Triangle-Diameter Conjecture. Given: HYP is inscribed in circle O such that HP is the diameter of the circle. Prove: HYP is a right triangle. Statements Reasons 694 Chapter Circles

63 Congratulations! You have just proven the Inscribed Right Triangle-Diameter Conjecture. It is now known as the Inscribed Right Triangle-Diameter Theorem. You can use this theorem as a valid reason in proofs. The Inscribed Right Triangle-Diameter Theorem states: If a triangle is inscribed in a circle such that one side of the triangle is a diameter of the circle, then the triangle is a right triangle. 7. Create a two-column proof of the Inscribed Right Triangle-Diameter Converse Conjecture. Y H O P R Given: Right HYP is inscribed in circle O. Prove: HP is the diameter of circle O. Statements Reasons Congratulations! You have just proven the Inscribed Right Triangle-Diameter Converse Conjecture. It is now known as the Inscribed Right Triangle-Diameter Converse Theorem. You can use this theorem as a valid reason in proofs. The Inscribed Right Triangle-Diameter Converse Theorem states: If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. A circumscribed polygon is a polygon drawn outside a circle such that each side of the polygon is tangent to a circle. Mr. Scalene asked his geometry class to draw a triangle and use a compass to draw a circle inside the triangle such that the circle was tangent to each side of the triangle. See if you perform this task. Lesson.6 Replacement for a Carpenter s Square 695

64 8. Use a straightedge to draw a triangle. a. Use your compass to draw a circle inside the triangle such that each side of the triangle is tangent to the circle. b. Were you able to do it? Explain. Rachel in Mr. Scalene s class did this task with no problem. Mr. Scalene asked Rachel to give others in the class a hint. She smiled and said, It has something to do with what we learned about a point of concurrency! c. Use your triangle to discover how the student was able to easily draw a circumscribed triangle and explain how you were able to do it. 696 Chapter Circles

65 PROBLEM 2 Quadrilaterals and Circles 1. Use your compass to draw a circle. a. Use your straightedge to draw an inscribed quadrilateral that is not a parallelogram in your circle. Label the vertices of your quadrilateral. b. Use your protractor to find the measures of the angles of the quadrilateral. What is the relationship between the measures of each pair of opposite angles? c. Write an Inscribed Quadrilateral-Opposite Angles Conjecture about the opposite angles of an inscribed quadrilateral. Let s prove the conjecture! Lesson.6 Replacement for a Carpenter s Square 697

66 2. Create a two-column proof of the Inscribed Quadrilateral-Opposite Angles Conjecture. U Q O A D Given: Quadrilateral QUAD is inscribed in circle O. Prove: Q and A are supplementary angles. U and D are supplementary angles. Statements Reasons Congratulations! You have just proven the Inscribed Quadrilateral-Opposite Angles Conjecture. It is now known as the Inscribed Quadrilateral-Opposite Angles Theorem. You can use this theorem as a valid reason in proofs. 698 Chapter Circles

67 The Inscribed Quadrilateral-Opposite Angles Theorem states: If a quadrilateral is inscribed in a circle, then the opposite angles are supplementary. Mrs. Rhombi asked her geometry class to draw a quadrilateral and use a compass to draw a circle inside the quadrilateral such that the circle was tangent to each side of the quadrilateral. See if you can do it. Most of her students used a straightedge and compass and drew this quadrilateral and tried to draw a circle inside the quadrilateral as shown. One of her students talked to a friend who happened to be in Mr. Scalene s class so she knew exactly what to do if it was a triangle. Will this information help her if it is a quadrilateral? Are you able to circumscribe the quadrilateral the same way you circumscribed the triangle? 3. Use the quadrilateral shown and try it for yourself. Explain your process. Lesson.6 Replacement for a Carpenter s Square 699

68 4. Ms. Rhombi then wrote the following theorem on the blackboard: A circle can be inscribed in a quadrilateral if and only if the angle bisectors of the four angles of the quadrilateral are concurrent. Using this theorem, how can you tell if it is possible to inscribe a circle in the quadrilateral in Question 3? Is it possible to inscribe a circle in this quadrilateral? Be prepared to share your solutions and methods. 700 Chapter Circles

69 .7 Gears Arc Length OBJECTIVE In this lesson you will: Determine the length of an arc of a circle. KEY TERM arc length Gears are used in many mechanical devices to provide torque, or the force that causes rotation. For instance, an electric screwdriver contains gears. The motor of an electric screwdriver can make the spinning components spin very fast, but the gears are needed to provide the force to push a screw into place. Gears can be very large or very small, depending on their application. Often gears work together, such as the gears shown. PROBLEM 1 Large and Small Gears Consider the circles shown that model two gears that work together. A B 1. Use your protractor to draw a central angle on each circle that has a measure of 60º. 2. What is the measure of each of the minor arcs associated with these central angles? Lesson.7 Gears 701

70 3. Write a fraction that compares the degree measure of the minor arc to the degree measure of the entire circle. 4. When gears are used together, the circumference of the gears are important because the gears move together. Describe the circumference of a circle. 5. Which gear in Problem 1 has a greater circumference? Explain your reasoning. 6. What fraction of the circumference do you think is taken up by the minor arcs described previously? Explain your reasoning. A portion of the circumference of a circle is called an arc length. 7. Suppose that the circumference of circle A is 48 inches and the circumference of circle B is 36 inches. What are the lengths of the minor arcs? Express your answer in terms of. 8. Consider the minor arcs of the central angles you drew. How do the measures of the arcs compare? How do the lengths of the arcs compare? 9. How is the measure of an arc different from the length of an arcs? Express your answer in terms of. 10. Use complete sentences to explain how you can calculate the length of an arc when you know its measure. a. Consider the circle shown. What is the circumference of the circle? Express your answers in terms of. r = 4 meters 702 Chapter Circles

71 b. Choose two points on the circle and label the points as point A and point B. Then, use a protractor to determine the measure of AB. Arc AB is what fraction of the circle? c. Use your answers to write an expression for the arc length of AB. Express your answers in terms of.. Write an expression to represent the length of AB in the circle shown. A r B 12. Calculate the arc length of each circle. Express your answer in terms of. a. c. 10 inches A 80 A b. 80 B B d. 10 inches A 120 A 120 B 20 inches 20 inches B Lesson.7 Gears 703

72 13. Describe the relationship between the measure of an arc and its arc length. 14. Two semicircular cuts were taken from the rectangular region shown. Determine the perimeter of the shaded region. Do not express your answer in terms of Chapter Circles

73 15. Use the diagram shown to answer each question. a. The radius of a small tree ring (small circle) is r and the radius of a larger tree ring (large circle) is 10r. How does the arc length of the minor arc in the small tree ring compare to the arc length of the minor arc in the large tree ring? b. If the arc length of the minor arc in the small tree ring is equal to 3 inches, what is the arc length of the minor arc in the large tree ring? c. If m A 20, the length of the radius of the small tree ring is r, the length of the radius of the large tree ring is 10r, and the length of the minor arc of the small tree ring is 3 inches, determine the circumference of the large tree ring. d. Did you have to know the length of the radius to determine the circumference? Lesson.7 Gears 705

74 16. Joan wanted to measure the circumference of the tree in her front yard. She wrapped a piece of rope around a tree trunk and measured the length of the rope. If the length of the rope was 4.5 feet, determine the length of the radius of the tree. Do not express your answer in terms of. Be prepared to share your solutions and methods. 706 Chapter Circles

75 .8 Playing Darts Sectors and Segments of a Circle OBJECTIVES In this lesson you will: Determine the area of sectors of a circle. Determine the area of segments of a circle. KEY TERMS sector of a circle segment of a circle Consider the following diagram of a standard dartboard. Each different section of the board is surrounded by wire and the numbers indicate scoring for a game. There are different games with different scoring that can be played on a dartboard, but the highest score from a single throw occurs when a dart lands at the very center, or bullseye, of the dartboard. Lesson.8 Playing Darts 707

76 PROBLEM 1 Hitting the Bullseye The dartboard is made of concentric circles which are two circles that have the same center, but different radii. 1. The first circle inside the outermost circle of the dartboard has a diameter of 170 millimeters. Calculate the area of this circle. Express your answer in terms of. 2. Imagine that the pie-shaped sections extend to the center of the circle. How many pie-shaped sections is this circle divided into? 3. What is the measure of the central angle formed by one of these pie-shaped sections if all of the sections are congruent? Explain your reasoning. 4. What is the measure of the minor arc associated with this central angle? 5. What fraction of the circle is the minor arc? 6. What fraction of the circle s area is covered by one of the pie-shaped sections? Explain your reasoning. 7. How do the fractions in Questions 6 and 7 compare? 8. What is the area of one pie-shaped section? Express your answer in terms of. 708 Chapter Circles

77 A sector of a circle is a region of the circle bounded by two radii and the included arc. 9. How can the sides of the pie-shaped section be described with respect to the circle? 10. Draw and shade a sector on the circle shown. Label the points that form the sector. Name the radii and the included arc that define the sector. O. Describe how to determine the area of a sector if the length of the radius and the measure of the included arc are known. 12. Use the figure shown to write an expression representing the area of the sector. A O r B 13. Use the figure to write an expression representing the arc length of the included arc. 14. Compare the two expressions. What is the same? What is different? Consider the dartboard in Problem 1. Suppose the innermost circle divided into 20 sectors and has a diameter of 108 millimeters. Notice that half of the sectors on the dartboard are the same color. Lesson.8 Playing Darts 709

78 15. Describe two methods for calculating the total area of all sectors of the same color. Then calculate this area and the area of one sector. Express your answers in terms of. A segment of a circle is a region of the circle bounded by a chord and the included arc. A C B 16. Name the chord and arc that bound the shaded segment of the circle. 17. Describe a method for calculating the area of the segment of the circle. 18. If the length of the radius of circle C is 8 centimeters and m ACB 90, use your method to determine the area of the shaded segment of the circle. Express your answer in terms of. Then rewrite your answer rounded to the nearest hundredth. 710 Chapter Circles

79 19. The area of the segment shown is 9 18 square feet. Calculate the radius of circle O. A O B 20. The area of the segment is square feet. Calculate the radius of circle O. Lesson.8 Playing Darts 7

80 21. The length of the radius is 10 inches. Calculate the area of the shaded region of circle O. Express your answer in terms of. A O B Be prepared to share your solutions and methods. 712 Chapter Circles

81 .9 OBJECTIVES In this lesson you will: The Coordinate Plane Circles and Polygons on the Coordinate Plane Apply theorems to circles in a coordinate plane. Classify polygons in the coordinate plane. Use midpoints to determine characteristics of polygons. Distinguish between showing something is true under certain conditions, and proving it is always true. This lesson provides you an opportunity to review several familiar theorems and classify polygons formed in a coordinate plane. Circles and polygons located in a coordinate plane enable you to calculate distances, slopes, and equations of lines. It is important to understand the difference between showing something is true under certain conditions and proving something is always true. You will experience and differentiate between both instances in the following problems. Lesson.9 The Coordinate Plane 713

82 PROBLEM 1 OP 2.5 units HY 4 units PY 3 units The figure shown is circle O with the center point at the origin. The length of the radius is 2.5 units. 1. Determine m HP H y O 2 4 Y 2 P x Use a compass and the grid shown to construct: all points 4 units from ( 2.5, 0) all points 3 units from (2.5, 0) y H ( 2.5, 0) O 2 Y P (2.5, 0) x Label the point at which both constructed circles intersect point Y Is HYP a right triangle? Justify your conclusion. 714 Chapter Circles

83 5. Where is point Y located with respect to circle O? 6. What kind of angle is HYP with respect to circle O? 7. Describe the arc intercepted by HYP. 8. Describe the chord determined by the intercepted arc. 9. Which side is the hypotenuse of the right triangle? 10. Is the hypotenuse of the right triangle also a diameter of the circle?. What theorem does this problem illustrate? 12. Is this an instance of showing something is true under certain conditions, or proving something is always true? Lesson.9 The Coordinate Plane 715

84 PROBLEM 2 Line AT and AN intersect at point A and are drawn tangent to circle O at points T and N, respectively. The coordinates of point T are (4, 3) and the coordinates of point N are (0, 5). The center of circle C is at the origin and the length of the radius is 5. y C (0, 0) 4 2 O 2 2 T (4, 3) x N (0, 5) A 1. To determine if the length of the tangent segments are equal, what additional information is needed? 2. Describe a strategy to determine the additional information needed. 716 Chapter Circles

85 3. Determine the slope of each radius. 4. Determine the slope of each tangent line. 5. Determine an equation for each tangent line. 6. Determine the coordinates of point A. Lesson.9 The Coordinate Plane 717

86 7. Show the lengths of the tangent segments are equal. 8. What theorem does this problem illustrate? 9. Is this an instance of showing something is true under certain conditions, or proving something is always true? PROBLEM 3 Line AT is tangent to circle C at point T. Secant AG intersects circle C at points N and G. Line AT intersects AG at point A. The coordinates of point A are (10, 0). The center of circle C is at the origin and the length of the radius is 6. y T 2 8 G 6 C (0, 0) 4 2 O N 8 x A (10, 0) Describe a strategy to show the product of the secant segment and the external secant segment is equal to the square of the length of the tangent segment. 718 Chapter Circles

87 2. Use your strategy to show this equality. 3. What theorem does this problem illustrate? 4. Is this an instance of showing something is true under certain conditions, or proving something is always true? Lesson.9 The Coordinate Plane 719

88 PROBLEM 4 1. Is the quadrilateral formed by connecting the midpoints of the sides of a square also a square? The coordinates on the diagram shown define a square so it is important to remember x y. y (0, y) (x, y) (0, 0) (x, 0) x 2. Is this an instance of showing something is true under certain conditions, or proving something is always true? 720 Chapter Circles

89 PROBLEM 5 1. Draw four points on the circle. Provide the coordinates and labels for each point. Connect the points to form a quadrilateral. Classify the polygon formed by connecting the midpoints of the sides of the quadrilateral. y O x Is this an instance of showing something is true under certain conditions, or proving something is always true? Lesson.9 The Coordinate Plane 721

90 PROBLEM 6 1. Is a rhombus formed by connecting the midpoints of the sides of an isosceles trapezoid? a. Draw an isosceles trapezoid. b. Inscribe the trapezoid in a circle. c. Label the vertices of the trapezoid. d. Choose reasonable coordinates for each vertex. e. Use the coordinates to determine if this conjecture could be true. y N 4 x Chapter Circles

91 2. Is this an instance of showing something is true under certain conditions, or proving something is always true? PROBLEM 7 Summary 1. Describe the difference between showing something is true under certain conditions and proving something to be true under all conditions. Be prepared to share your solutions and methods. Lesson.9 The Coordinate Plane 723

92 Chapter Checklist KEY TERMS center of a circle (.1) chord (.1) minor arc (.1) semicircle (.1) external secant segment (.5) secant of a circle (.1) tangent of a circle (.1) degree measure (of an arc) (.2) inscribed polygon (.6) circumscribed polygon (.6) point of tangency (.1) adjacent arcs (.2) arc length (.7) central angle (.1) intercepted arc (.2) sector of a circle (.8) inscribed angle (.1) segments of a chord (.4) segment of a circle (.8) arc (.1) tangent segment (.5) major arc (.1) secant segment (.5) POSTULATE Arc Addition Postulate (.2) THEOREMS Inscribed Angle Theorem (.2) Parallel Lines-Congruent Arcs Theorem (.2) Interior Angles of a Circle Theorem (.3) Exterior Angles of a Circle Theorem (.3) Diameter-Chord Theorem (.4) Equidistant Chord Theorem (.4) CONSTRUCTION circumscribed polygons (.6) Equidistant Chord Converse Theorem (.4) Congruent Chord-Congruent Arc Theorem (.4) Congruent Chord-Congruent Arc Converse Theorem (.4) Segment-Chord Theorem (.4) Tangent Segment Theorem (.5) Secant Segment Theorem (.5) Secant Tangent Theorem (.5) Inscribed Right Triangle- Diameter Theorem (.6) Inscribed Right Triangle- Diameter Converse Theorem (.6) Inscribed Quadrilateral- Opposite Angles Theorem (.6) 724 Chapter Circles