b. Draw a line and a circle that intersect at exactly one point. When this happens, the line is called a tangent.

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1 6-1. Circles can be folded to create many different shapes. Today, you will work with a circle and use properties of other shapes to develop a three-dimensional shape. Be sure to have reasons for each conclusion you make as you work. Start by drawing a large circle on paper with a compass and cutting it out. a. Fold the circle in half to create a crease that goes through the center of the circle. Unfold the circle and then fold it in half again to create a new crease that is perpendicular to the first crease. Unfold your paper back to the full circle, what is another name for the line segment represented by each crease? Explain why. b. On the circle, label the endpoints of one diameter A and B. Fold the circle so that point A touches the center of the circle and create a new crease. Then label the endpoints of this new crease C and D. What appears to be the relationship between segments AB and CD? Is CD a diameter? (why or why not). What is CD called? C A B D c. Now fold the circle twice to form creases BC and BD and use scissors to cut out ΔBCD. What type of triangle is ΔBCD? How can you be sure? Explain your reasoning.

2 d. Your equilateral triangle ΔBCD should now be flat (also called twodimensional). Two-dimensional shapes have length and width, but not depth (or thickness ). If you cut the labels off when creating your equilateral triangle, label the vertices of ΔBCD again. Then, with the unmarked side of the triangle facedown, fold and crease the triangle so that B touches the midpoint of CD. Keep it in the folded position. What does the resulting shape appear to be? What smaller shapes do you see inside the larger shape? Justify that your ideas are correct. fold e. Open your shape again so that you have the large equilateral triangle in front of you. How does the length of a side of the large triangle compare to the length of the side of the small triangle formed by the crease? How many of the small triangles would fit inside the large triangle? In what ways are the small and large triangles related? f. Repeat the fold in part (d) so that C touches the midpoint of BD. Unfold the triangle and fold again so that D touches the midpoint of BC. Create a three-dimensional shape by bringing points B, C, and D together. A three-dimensional shape has length, width, and depth. Use tape to hold your shape together. g. Three-dimensional shapes formed with polygons have faces and edges, as well as vertices (plural of vertex). Faces are the flat surfaces of the shape, while edges are the line segments formed when two faces meet. Vertices are the points where edges intersect. How many faces, edges, and vertices does your three-dimensional shape have?

3 6-2. Consider all the ways a circle and a line can intersect. Can you visualize a line and a circle that intersect at exactly one point? What about a line that intersects a circle twice? On your paper, draw a diagram for each of the situations below, if possible. If it is not possible, explain why. a. Draw a line and a circle that do not intersect. b. Draw a line and a circle that intersect at exactly one point. When this happens, the line is called a tangent. c. Draw a line and a circle that intersect at exactly two points. A line that intersects a circle twice is called a secant. d. Draw a line and a circle that intersect three times A line that intersects a circle exactly once is called a tangent. What is the relationship of a tangent to a circle? To investigate this question, carefully copy the diagram showing line l tangent to A below onto your paper. Fold the paper so that the crease is perpendicular to line l through point P. Your crease should pass through point C. What does this tell you about the tangent line? 6-4. Use the relationships in the diagram below to answer the following question. Be sure to name what relationship you used. PQ is tangent to C at P. If PQ = 5 and CQ = 6, find CP, m C, and m PQC. ex 6-4 ex In the figure above, EX is tangent to O at point X. OE = 20 cm and XE = 15 cm. Find OX and EN.

4 6-6. Ventura began to think about perpendicularity in a circle. He wondered, If a radius is perpendicular to a line at a point on the circle, how do we know if that the line is a secant or a tangent?. a. Miguel says, Let s assume the line perpendicular to the radius is a secant. On your paper, draw a diagram, like the one below, with C and a secant. Label the points where the secant intersects the circle A and B. Since Ventura s question assumes that the line is perpendicular to a radius, assume that AB is perpendicular to CA at A. b. Ventura adds I think ΔCAB is isosceles. Do you agree? Explain how you know. c. Sandra chimes in with, Then CBA must be a right angle too. Ventura quickly adds, But that s impossible! What do you think? Give reasons to support your conclusions. d. Explain to Ventura what this contradiction reveals about the line perpendicular to the radius of a circle at a point on the circle In the diagram below, M has radius 14 feet and A has radius 8 feet. ER is tangent to both M and A. If NC = 17 feet, find ER. Copy the diagram. a. First find the length of MA. Then subtract the radii to get MB. b. Use Pythagorean theorem to get AB. Why is ER = AB? Explain.

5 6-8. In the diagram below, AD is a diameter of ΘB. a.if m A = 35, what is m ABC and m C? b.if m CBD = 68, what is m ABC and m C? c.if m A = 15, copy the diagram and find all angles and arcs? 6-9. Leticia started a construction below. Explain what she is constructing. Then copy her diagram and finish her construction For each of the geometric relationships represented below, write and solve an equation for the given variable. For part (a) assume that C is the center of the circle. Show all work On graph paper, plot ΔABC if A( 1, 1), B(1, 9) and C(7, 5). a. Find the midpoint of AB and label it D. Also find the midpoint of BC and label it E. b. Find the length of the midsegment, DE. Use it to predict the length of AC. c. Now find the length of AC and compare it to your prediction from (b).

6 6-12. Assume point B is the center of the circle below. Give the best description name for each part. a. AB b. CD c. AD d. ACD e. ABC f. AC g. ADC For each pair of triangles below, decide if the triangles are similar or not and explain how you know. If the triangles are similar, complete the similarity statement ΔABC Δ. a. b For each triangle below, solve for the given variables a. b. c.

7 6-15.The baobab tree is a species of tree found in Africa and Australia. It is often referred to as the world s widest tree because it is up to 45 feet in diameter. A B While digging at an archeological site, Rafi found a fragment of a fossilized baobab tree that appears to be wider than any tree on record! However, since he does not have the remains of the entire tree, he cannot simply measure across the tree to find its diameter. Assume that the shape of the tree s cross-section is a circle. The arc AB above is a scaled representation of Rafi s tree fragment. a. Trace this arc as neatly as possible on paper. One way to find the center of a circle when given an arc is to fold it so that the two ends of the arc coincide (lie on top of each other). If you fold AB so that A lies on B, what is the relationship between the resulting crease and the arc AB? Explain how you know. A B b. Now fold point A to where the crease bisects AB. The two creases will intersect at the center of the circle. Sketch the complete circular cross-section of the tree. A B c. If 1 cm represents 10 feet of tree, find the approximate lengths of the radius and diameter of the tree. Does the tree appear to be larger than 45 feet in diameter?

8 6-16. Examine the chord WX in Z below. If WX = 8 and the radius of Z is 5, how far from the center is the chord? Draw the diagram on your paper and show all work Examine the diagram of chord LM in P below. If the length of the radius of P is 5 units and if LM is 3 units from the center, find LM. Show all your work In C below,. Prove that ACB DCE. Complete the flowchart proof In Y below, assume that. Prove that. Complete the flowchart proof.

9 6-20. In the circle below, F, G, and H are examples of inscribed angles. Notice that all three angles intercept the same arc (JK). Compare their measures. What do you notice? Now compare the measurements of the central angle (such as WZY in Z at below) and an inscribed angle (such as WXY). What is the relationship of an inscribed angle and its corresponding central angle? Use a protractor to measure the angles to test your idea Use the relationships in the diagrams to solve for x. Justify your solutions For ABCD inscribed in the circle below, solve for x. Explain how you found your answer.

10 6-24. In problem 6-21, you found that the measure of an inscribed angle was half of the measure of its corresponding central angle. Examine the diagrams below. Find the measures of the indicated angles. If a point is labeled C, assume it is the center of the circle For each diagram below, write an equation to represent the relationship between x and y UV is a diameter of the circle below. If TU = 6 and TV = 8, what is the diameter of the circle? What is its perimeter of the triangle?

11 6-27. Timothy asks, What if two chords intersect inside a circle? Can triangles help me learn something about these chords? Copy his diagram below in which chords and intersect at point E. a. Timothy decided to create two triangles (ΔBED and ΔACE). Add line segments and to your diagram. b. Compare B and C. What do you notice? Likewise, compare D and A. Write down your observations. c. How are ΔBED and ΔACE related? Justify your answer. d. If DE = 8, AE = 4, and EB = 6, then what is EC? Use proportions, show your work Copy the diagrams and solve for x. Justify your solutions. a. KL = 8 inches b. AE = 10, CE = 4, AB = 16, and DE = x In A below, is a diameter and m C = 64. Find: a. m D b. m BF c. m E d. m CBF e. m BAF f. m BAC

12 6-30. Solve for the variables in each of the diagrams below. Show all work Solve for the given variables in the diagrams below. Show all work If QS is a diameter and PO is a chord of the circle below, find the measure of the geometric parts listed below. a. m QO b. m QSO c. m QPO d. m ONS e. m PQS f. m PS g. m PQ

13 6-33. What type of equation could represent a circle? On a piece of graph paper, draw a set of xy-axes. Then use a compass to construct a circle with radius of length 10 units centered at the origin (0, 0). a. Find all of the points on the circle where x = 6. For each point, what is the y-value? Use a right triangle (like the one shown at right) to justify your answer. b. What if x = 3? For each point on the circle where x = 3, find the corresponding y- value. Use a right triangle to justify your answer. c. Mia picked a random point on the circle and labeled it (x, y). Unfortunately, she does not know the value of x or y! Help her write an equation that relates x, y, and 10 based on her diagram above. Use Pythagorean theorem. d. Does your equation from part (d) work for the points (10, 0) and (0, 10)? What about ( 8, 6)? Explain In problem 6-33, you wrote an equation of a circle with radius of length 10 units and center at (0, 0). a. What if the radius were instead 4 units long? write an equation of this circle. b. Write the equation of a circle centered at (0, 0) with radius r. c. On graph paper, sketch the graph of x 2 + y 2 = 36. Can you graph it without a table? Explain your method Review circle relationships as you answer the questions below. a. On your paper, draw a diagram of B with. If = 90 and the length of the radius of B is 10, find the length of chord. b. Now draw a diagram of a circle with two chords, EF and GH, that intersect at point K. If EF = 15, EK = 6, and HK = 3, what is GK?

14 6-36. What if the center of the circle is not at (0, 0)? On graph paper, construct a circle with a center A(3, 1) and radius of length 5 units. a. On the diagram above, point P represents a point on the circle with no special characteristics. Add a point P to your diagram and then draw a right triangle like ΔABP in the circle at right. b. What is the length PB? Write an expression to represent this length. Likewise, what is the length AB? c. Use your expressions for AB and PB, along with the fact that the length of the radius of the circle is 5 units, to write an equation for this circle. (Note: You do not need to worry about multiplying any binomials.) d. Find the equation of each circle represented below. (1) The circle with center (2, 7) and radius of length 1 unit. (3) The circle for which (6, 0) and ( 6, 0) are the endpoints of a diameter In the diagram below, AB is a diameter of L. If BC = 5 and AC = 12, use the relationships shown in the diagram to solve for the quantities listed below. a. AB b. AL c. m B d. m AC e. m BC

15 6-38. As Ventura was doodling with his compass, he drew the diagram below. Assume that each circle passes through the center of the other circle. a. Explain why A and B must have the same radius. b. Construct two intersecting circles so that each passes through the other s center. Label the centers A and B, as above. c. On your construction, locate the two points where the circles intersect each other. Label these points C and D. Then draw quadrilateral ACBD. What type of quadrilateral is ACBD? Justify your answer. d. Use what you know about the diagonals of ACBD to describe the relationship of AB and CD. Make as many statements as you can Copy diagrams below and answer the following questions. Show all work. a. In H, = 40 and = 210. Find,, and m RGO. b. ΔABC is inscribed in the circle below. Using the measurements provided in the diagram, find, m AC, m AB, m A, and m C.

16 6-40. Solve for the variables in each of the diagrams below. Show all work Copy the diagram below onto your paper. Assume AD is tangent to C at D. a. If AD = 9 and AB = 15, what is the diameter DB and radius DC? b. If m ED = 60, what is m EB, m B, m EDB? c. If = 120 and if BC = 6, find ED and EB.