CHAPTER 10 PROPERTIES OF CIRCLES

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1 HT 0 OTIS OF ILS In this chapter we address ig IS: ) Using properties of segments that intersect circles ) pplying angle relationships in circles 3) Using circles in the coordinate plane Section: ssential Question 0 Use roperties of Tangents How can you verify that a segment is tangent to a circle? Warm Up: Key Vocab: ircle The set of all points in a plane that are equidistant from a given point called the center of the circle. enter The point from which all points of the circle are equidistant oint is the center of segment whose endpoints are the center of the circle and a point on the circle. adius The distance from the center of a circle to any point on the circle. lural: radii is a radius Student Notes Geometry hapter 0 roperties of ircles KY age #

2 X Y hord segment whose endpoints are on the circle. W Z WX and YZ are chords. iameter chord that passes through the center of the circle. The distance across a circle through its center. is a diameter Secant line that intersects a circle in two points. m Line m is a secant. S Tangent line in the plane of a circle that intersects the circle in exactly one point, the point of tangency. Q QS is tangent to at point of tangency. Student Notes Geometry hapter 0 roperties of ircles KY age #

3 Theorems: In a plane, a line is tangent to a circle IFF the line is perpendicular to a radius of the circle at its endpoint on the circle Q m line m is tangent to Q, m Q m Q line m is tangent to Q. Tangent segments are drawn from a common external point, They are congruent. S and TS are tangents, S TS S S T T Show: x : Tell whether the line or segment is best described as a radius chord, diameter, secant, or tangent of. F a.) radius c.) diameter b.) chord d.) tangent Student Notes Geometry hapter 0 roperties of ircles KY age #3

4 x : Tell how many common tangents the circles have and draw them. a.) b.) c.) external tangents 4 common tangents external, internal 3 common tangents external, internal x 3: In the diagram, is a radius of. Is tangent to? xplain No, , and x 4: In the diagram, S is a point of tangency. Find the radius r of T. r T S r 48 cm 36 cm r 36 r 48 r 7r 96 r 304 7r 08 r 4 cm x 5: In, is tangent at and is tangent at. Find x. 5 6x-8 5 6x x 5.5 x Student Notes Geometry hapter 0 roperties of ircles KY age #4

5 Section: ssential Question 0 Find rc Measures How do you find the measure of an arc of a circle? Warm Up: Key Vocab: entral ngle n angle whose vertex is the center of the circle. rc portion of the circumference of the circle. is a central angle of Minor rc art of a circle that measures less than 80 Major rc art of a circle that measures greater than 80 is a minor arc is a major arc Student Notes Geometry hapter 0 roperties of ircles KY age #5

6 Semicircle n arc with endpoints that are the endpoints of a diameter of a circle. The measure of a semicircle is 80 F F is a semicircle. 70 Measure of an rc The measure of the arc s central angle m 70 m ongruent ircles ircles that have congruent radii Q Q ongruent rcs Two arcs that have the same measure and are arcs of the same circle or of congruent circles F F ostulates: rc ddition ostulate The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. is between and, The sum of the parts equals the whole then m m m then m m m, is between and. Student Notes Geometry hapter 0 roperties of ircles KY age #6

7 Show: x : Find the measure of each arc of, where is a diameter a.) m 35 b.) m 5 c.) m x : result of a survey about the ages of people in a town is shown. Find the indicated arc measures. a.) mu 40 b.) mst =30 c.) d.) mvt = 30 must = 70 x 3: Tell whether arcs and F are congruent. xplain why or why not. a.) b.) c.) F F 0 Q F F F ; they are in the same circle and m mf. and F have the same measure, but they are not congruent because they are arcs of circles that are not congruent. F because they are in the same circle and vertical angles and F are congruent. Student Notes Geometry hapter 0 roperties of ircles KY age #7

8 Section: ssential Question 0 3 pply roperties of hords How can you tell if two chords in a circle are congruent? Warm Up: Theorems: In the same circle or in congruent circles, two minor arcs are congruent IFF their corresponding chords are congruent.,. Student Notes Geometry hapter 0 roperties of ircles KY age #8

9 one chord is a perpendicular bisector of another chord, Q is perp. bis. of ST, S the first chord is a diameter. Q is a diameter S Q U Q U a diameter of a circle is perpendicular to a chord, T the diameter bisects the chord N its arc. T Q is a diameter and Q S ST, SU TU and S T. S Q U Q U T T In the same circle or in congruent circles, two chords are congruent IFF they are equidistant from the center. G F, G FG G FG. Student Notes Geometry hapter 0 roperties of ircles KY age #9

10 Show: x : In, and m 08. Find m. 08 m 08 x : Three props are place on a stage at points, Q, and as shown. escribe how to find the location of a table so it is the same distance from each prop. Q raw the perpendicular bisectors of any two of Q, Q, and. The intersection of the two perpendicular bisectors is the desired location. x 3: Use the diagram of 0 F to find the length of F. Tell what theorem your used. F 0 a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. G x 4: In the diagram of, V W, Q x 6, and ST 3x. Find Q. V S W T x 6 3x 7 x Q (7) 6 Q 0 Q Student Notes Geometry hapter 0 roperties of ircles KY age #0

11 Section: 0 4 Use Inscribed ngles and olygons ssential Question How do you find the measure of an inscribed angle? Warm Up: Key Vocab: Inscribed ngle Intercepted rc n angle whose vertex is on a circle and whose sides contain chords of the circle. The measure of an inscribed angle is one half the measure of its intercepted arc. The arc that lies in the interior of an inscribed angle and has endpoints on the angle. is an inscribed angle. is its intercepted arc. m m Inscribed olygon polygon whose vertices all lie on a circle. ircumscribed ircle The circle that contains the vertices of an inscribed polygon. The quadrilateral and the triangle are inscribed in the circles. The circles are circumscribed about the quadrilateral and the triangle. Student Notes Geometry hapter 0 roperties of ircles KY age #

12 Theorems: two inscribed angles of a circle intercept the same arc, Inscribed angles and both intercept, the angles are congruent. right triangle can be inscribed in a circle IFF the hypotenuse is a diameter of the circle. in, m 90, is a diameter of the circle is a diameter of the circle in, m 90. quadrilateral can be inscribed in a circle IFF its opposite angles are supplementary F 74 G,,F, and G lie on the circle, m mf 80 mg m m mf 80 mg m,,f, and G lie on the circle. Student Notes Geometry hapter 0 roperties of ircles KY age #

13 Show: x : Find the indicated measure in. a.) m b.) m 0 m m x : Find mkn and m KMN. What do you notice about KMN and KLN? K 5 L M mkn (5) 04 mkmn 5 KMN KLN N x 3: Name two pair of congruent angles in the figure. S.) S U T.) T U S U T.) U S T.) T S x 4: graphic design software program was used in a home improvement store to design kitchen cabinets. The designer showed a wall of cabinets with a90viewing angle at. From what other positions would the cabinets fill a90viewing window? From any position on a semicircle that has as a diameter. Student Notes Geometry hapter 0 roperties of ircles KY age #3

14 x 5: Find the value of each variable. a.) b.) 00 y 7y 7x 60 x 5x 9y 00 x80 60 y80 x80 y0 5x 7x 80 7 y 9y 80 x80 36y 80 x5 y 5 Student Notes Geometry hapter 0 roperties of ircles KY age #4

15 Section: ssential Question 0 5 pply Other ngle elationships in ircles How do you find the measure of an angle formed by two chords that intersect inside a circle? Warm Up: Theorems: a tangent and a chord intersect at a point on a circle, the measure of each angle formed is one half the measure of its intercepted arc. m m and m m Student Notes Geometry hapter 0 roperties of ircles KY age #5

16 two chords intersect inside a circle, ngles Inside the ircle Theorem then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.. m m m m m m and ngles Outside the ircle Theorem a tangent and a secant, two tangents, or two secants intersect outside a circle,, the measure of the angle formed is one half the difference of the measures of the intercepted arcs a tangent and a secant two tangents two secants m Q S V U mqs ms m mtu m 3 mzy mwx T mtvu 3 W X Y Z Student Notes Geometry hapter 0 roperties of ircles KY age #6

17 Show: x : Line m is tangent to the circle. Find the indicated variable. a.) b.) 8 x m m 8 y x (8) 4 8 y 36 y x : Find the value of x. x 89 x x 3: Find the value of x x x x 4: The Space Shuttle typically orbits at a height of 600 kilometers. Suppose an astronaut takes a picture from point S. What is the measure of Q? (ound your answer to the nearest tenth) Student Notes Geometry hapter 0 roperties of ircles KY age #7

18 6370 ms sin msq ms msq 360 x x x x x x x Student Notes Geometry hapter 0 roperties of ircles KY age #8

19 Section: ssential Question 0 6 Find Segment Lengths in ircles What are some properties of chords, secants, and tangents to a circle? Warm Up: Key Vocab: Y Segments of a hord When two chords intersect in the interior of a circle, each chord is divided into two segments called segments of the chord. WU XU W X U and UZ are segments of chord WZ. and UY are segments of chord XY. Z Secant Segment segment that contains a chord of a circle and has exactly one endpoint outside the circle. xternal Segment The part of a secant segment that is outside the circle is a secant segment. is the external segment. Student Notes Geometry hapter 0 roperties of ircles KY age #9

20 Theorems: two chords intersect in the interior of a circle, Segments of hords Theorem 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 other chord. two secant segments share the same endpoint outside a circle, Segment of Secants Theorem the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment. Segments of Secants and Tangents Theorem a secant segment and a tangent segments share an endpoint outside a circle, the product of the lengths of the secant segment and its external segment equals the square of the length of the tangent segment. Student Notes Geometry hapter 0 roperties of ircles KY age #0

21 Show: x : Find T and SU. U x+ 4x V x S 3x T 3x 3x x x4x 0 x 3x4x T (3) 3(3) 3 3x 0 x x 3 x 0, 3 SU ( 3) x : What is the value of x? 3 x 4 x x 4 3x 5 x 5.).) 3.) 5.) 8 x 3: Use the figure to find. x x x x 0 x Student Notes Geometry hapter 0 roperties of ircles KY age #

22 x 4: Suppose that Space Shuttle is positioned at point S, 50 miles above the arth. What is the length of line segment S from the arth to the shuttle? (The radius of the arth is about 4000 miles.) ound your answer to the nearest mile. S S S 06 miles Student Notes Geometry hapter 0 roperties of ircles KY age #

23 Section: ssential Question 0 7 Write and Graph quations of ircles What do you need to know to write the standard equation of a circle? Warm Up: Key Vocab: Standard quation of a ircle The standard equation of a circle with center ( hk, ) and radius r is ( xh) ( y k) r rea of a ircle r Sector of a ircle region of a circle that is bounded by two radii and an arc of the circle. m rea of a Sector r 360 Student Notes Geometry hapter 0 roperties of ircles KY age #3

24 Show: x : Write the equation of the circle shown. y 3 x y x x : Write the standard equation of a circle with center (,3) and radius 3.8. x y x 3: The point (8, ) is on a circle with center (4,). Write the standard equation of the circle. x4 y 5 x 4: The equation of a circle is x y 3 4. Graph the circle. 6 y x Student Notes Geometry hapter 0 roperties of ircles KY age #4

25 x 5: In O with radius 9, mo 0. Find the area of the circle and the areas of each of the sectors. circle circle sector sector sector sector x 6: Find the area of the region bounded by and sector O 9 3. total 0 6 O 6 Student Notes Geometry hapter 0 roperties of ircles KY age #5