NOTES: Tangents to Circles

Transcription

1 Unit# ssign # TS: Tangents to ircles GL Identify segments and lines related to circles and use properties of a tangent to a circle VULRY circle is the set of all points in a plane that are equidistant from a given point, called the center of the circle. The distance from the center to a point on the circle is the radius of the circle. Two circles are congruent if they have the same radius. The distance across the circle, through its center, is the diameter of the circle. radius is a segment whose endpoints are the center of the circle and a point on the circle. chord is a segment whose endpoints are points on the circle. diameter is a chord that passes through the center of the circle. secant is a line that intersects a circle in two points. tangent is a line in the plane of a circle that intersects the circle in eactly one point. Theorem 0. If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. Theorem 0. In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle. Theorem 0.3 If two segments from the same eterior point are tangent to a circle, then they are congruent. Using roperties of Tangents and are tangent to. ind the value of. SLUTI Two tangent segments from the same point are congruent. Substitute. dd 3 to each side Subtract 5 from each side.

2 LTUR: Tangents to ircles Unit# ssign # The diameter of a circle is given. ind the radius.. d 6 in.. d 4 cm 3. d 5 ft 4. The radius of a circle is given. ind the diameter. 5. r cm 6. r 8 ft 7. r 0 in. 8. d 9 in. r 4.6 cm atch the notation with the term that best describes it. 9.. enter 0. H. hord.. iameter.. Radius 3.. oint of tangency 4.. ommon eternal tangent 5. G. ommon internal tangent 6. H. Secant H G Use the diagram at the right. 7. What are the center and radius of? y 8. What are the center and radius of? 9. escribe the intersection of the two circles. 0. escribe all the common tangents of the two circles.. re the two circles congruent? plain. Tell whether is tangent to. plain your reasoning aseball Stadium The shape of the outfield fence in a baseball stadium is that of a quarter circle. If the distance from home plate to the wall is 330 feet, what is the radius of the entire circle? What is the diameter of the circle? 330 ft

3 lasswork: Tangents to ircles Unit# ssign # The diameter of a circle is given. ind the radius.. d 3 in.. d 8 cm 3. d.6 ft 4. The radius of a circle is given. ind the diameter. 5. r 7 cm 6. r 6.3 ft 7. r 0.75 in. 8. d ft 5 in. r 4.5 ft atch the notation with the term that best describes it. 9.. enter 0.. hord. HG. iameter.. Radius G H 3.. oint of tangency 4.. ommon eternal tangent 5. G. ommon internal tangent 6. G H. Secant Use the diagram at the right. 7. What are the center and radius of? 8. What are the center and radius of? 9. escribe the intersection of the two circles. 0. escribe all the common tangents of the two circles. y Tell whether is tangent to. plain your reasoning and are tangent to. ind the value of

4 Unit# ssign # Homework: Tangents to ircles ame the term that best describes the notation HG G H G opy the diagram. Tell how many common tangents the circles have. Then sketch the tangents or each find the value of. ssume that segments that appear to be tangent are tangent Use the diagram at the right. y 8. ind the radius of. t 9. ind the slope of line t. (3, ) (5, )

5 TS: rcs and hords Unit# ssign # GL Use properties of arcs of circles and use properties of chords of circles VULRY In a plane, an angle whose verte is the center of a circle is a central angle of the circle. If the measure of a central angle,, is less than 80, then and and the points of in the interior of form a minor arc of the circle. The points and and the points of in the eterior of form a major arc of the circle. If the endpoints of an arc are the endpoints of a diameter, then the arc is a semicircle. The measure of a minor arc is defined to be the measure of its central angle. The measure of a major arc is defined as the difference between 360 and the measure of its associated minor arc. Two arcs of the same circle or of congruent circles are congruent arcs if they have the same measure. ostulate 6 rc ddition ostulate The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. Theorem 0.4 In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. Theorem 0.5 If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. Theorem 0.6 If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. Theorem 0.7 In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. Using Theorem 0.7 S, TV, and SQ 7. ind QU. SLUTI ecause S TV, they are equidistant from the center by Theorem 0.7. To find QU, first find QR. QR S, so QR bisects S. ecause T R U Q 7 S V S, RS 6. ow look at QRS which is a right triangle. Use the ythagorean Theorem to find QR. QR QS RS ecause QR QU, QR QU 3.

6 Unit# ssign # rcs and hords etermine whether the arc is a minor arc, a major arc, or a semicircle of Q and are diameters. ind the indicated measure. 9. m R 0. mq 7. mr 8. mq ind the measure of.. mqr. 3. mqr mr 4. mr 5. mqr 6. mq Q R Q What can you conclude about the diagram? State a postulate or theorem that justifies your answer ind the indicated measure for. 5.? 6.? 7.? 4 8 6

7 Homework: rcs and hords Unit# ssign # etermine whether the arc is a minor arc, a major arc, or a semicircle of Q and are diameters. ind the indicated measure. 9. m R 0. mq. mqr. 3. mqr 4 5. mqr mr mr 6. mq 7. mr 8. mq ind the measure of Q R T T T 85 Q Q 30 Use the figures to match the chord or arc with a congruent chord or arc ind the indicated measure for. 8.? 9. 4 m 50,, m? G

8 Unit# ssign # tra redit: rcs and hords Q and are diameters. ind the indicated measures.. m R. mq ind the measure of. 3. mqr m mr mqr mq 8. mr 9. mqr 0. mq. mr. mq 8 73 Q R 3. R R 3 is the center of the circle. Use the given information to find XY. plain your reasoning. 6. ZY 3 7. ZY 6, XW Z Y X 3 Y X W Z V 3 X Y Write a two-column proof or a paragraph proof. 9. Given:, G 0. Given: G is a diameter of., Q, Q rove: rove: Q Q G

9 TS: Inscribed ircles Unit# ssign # GL Use inscribed angles to solve problems and use properties of inscribed polygons VULRY n inscribed angle is an angle whose verte is on a circle and whose sides contain chords of the circle. The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the intercepted arc of the angle. If all of the vertices of a polygon lie on a circle, the polygon is inscribed in the circle and the circle is circumscribed about the polygon. Theorem 0.8 easure of an Inscribed ngle If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc. Theorem 0.9 If two inscribed angles of a circle intercept the same arc, then the angles are congruent. Theorem 0. quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. inding the easure of an ngle If is a right angle, what is the measure of? SLUTI y Theorem 0.9, because the two angles intercept the same arc. So, m 90. Using an Inscribed Quadrilateral ind the value of each variable. SLUTI y y Theorem 0., the opposite angles of this quadrilateral are supplementary. So you can write the following equations and then solve for the variable in each y y 80 3y 80 y 60 y

10 Inscribed ircles Unit# ssign # ind the measure of the indicated arc or angle.. m?. m? 3. m? m? 5. m? 6. m? 3 4 ind the measure of the arc or angle in 7. mq 8. m m 0.. mq. 3. mq 4. mq m mq Q 70 ecide whether a circle can be circumscribed about the quadrilateral ind the value of each variable y y

11 Unit# ssign # Homework: Inscribed ircles ind the measure of the indicated arc or angle in.. m. m?? 3. m? ind the measure of the arc or angle in m 08 and m 00., given 4. m 5. m 6. m 7. m 8. m m. m m ind the value of diameter 96 ( ) ( 7) ( ) 5. ( 3) 6. (3 8) 7. ( ) (3) (5 ) ( + 4 ) 8. rcheology rcheologists found a portion of a circular dinner plate. escribe a method to determine the diameter of the plate.

12 Unit# ssign # tra redit: Inscribed ircles ind the measure of the indicated arc or angle in.. m. m?? 3. m? m? 5. m? 6. m? ind the measure of the arc or angle in m 86 and m 95., given 7. m m 0.. m. 3. m 4. m m m m ind the value of ( ) 7. center (5 4) ( ) 08 (4 3) Write a two-column proof or a paragraph proof. 8. Given: G rove: I ~ G ( + ) I

13 Unit# ssign # TS: ngle Relationships in ircles GL Use angles formed by tangents and chords to solve problems in geometry and use angles formed by lines that intersect a circle to solve problems VULRY XL Theorem 0. If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc. Theorem 0.3 If two chords intersect in the interior of a circle, 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. Theorem 0.4 If a tangent and a secant, two tangents, or two secants intersect in the eterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs. inding ngle and rc easures Line m is tangent to the circle. a. ind b. m m m 90 SLUTI a. m XL Using Theorem 0.3 b. ecause and are a linear pair, m 80 m So, m ind the value of. SLUTI 0 m m pply Theorem Substitute. Simplify.

14 Unit# ssign # ngle Relationships in ircles ind the measure of Write an equation that can be used to solve for. Then solve the equation for Use the diagram of to write the m, m, and m3 in order of increasing measure

15 Homework: ngle Relationships in ircles Unit# ssign # ind the measure of Write an equation that can be used to solve for. Then solve the equation for erial View You are flying across the plains of Kansas at an altitude of 3,000 feet, or approimately 6 miles. It is a clear day. ind the measure of that represents the part of arth that you can see mi 4006 mi not drawn to scale

16 Unit# ssign # tra redit: ngle Relationships in ircles ind the measure of Use the given information to solve the following. 7. Given: 8. Given: The two circles intersect at and ; m ind: m m 00 mx 70, 0, m 60 ind: The difference between the measures of of the smaller circle and of the larger circle X Write a two-column proof or a paragraph proof. 9. Given: is the midpoint of 0. Given: rove: bisects rove: ;

17 quations of ircles Unit# ssign # atch the equation of a circle with its description.. y 4. center, 4, radius 4. y 9. center, 3, radius 3 3. y 4 6. center 0, 0, radius 4. y 3 9. center, 5, radius y 5 6. center 3, 5, radius 4 6. y 5 9. center 0, 0, radius 3 Give the center and radius of the circle. 7. y y y 4 6. y 4 9 y 4 4 y 5 Give the coordinates of the center, the radius, and the equation of the circle y y y Write the standard equation of the circle with the given center and radius. 6. center 0, 0, radius 7. center 0,, radius 8. center, 0, radius 3 9. center 3, 3, radius 4 The equation of a circle is y 4. Tell whether each point is on the circle, in the interior of the circle, or in the eterior of the circle. 0.,., 4., , 4. 4, ,

18 Unit# ssign # Homework: quations of ircles atch the equation of a circle with its description.. y 3 4. center 3, 5, radius 4. y 5 9. center, 3, radius y 5 6. center, 3, radius 4. y 3 9. center, 5, radius y 5 6. center 3, 5, radius 4 6. y 5 9. center, 5, radius 3 Give the center and radius of the circle y y y y y y 4 6 Give the coordinates of the center, the radius, and the equation of the circle y 4 4 y y Write the standard equation of the circle with the given center and radius. 6. center 0, 0, radius 7. center 0, 4, radius 4 8. center 4,, radius 3 9. center 3, 5, radius 5 The equation of a circle is 4 y 9. Tell whether each point is on the circle, in the interior of the circle, or in the eterior of the circle. 0. 5,. 8,., 3. 4, , 5. 4,