Finding Angle Measures. Solve. 2.4 in. Label the diagram. Draw AE parallel to BC. Simplify. Use a calculator to find the square root. 14 in.


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1 lan bjectives 1 To use the relationship between a radius and a tangent To use the relationship between two tangents from one point amples 1 inding ngle Measures RealWorld onnection inding a Tangent Using Theorems 1 ircles Inscribed in olygons Math ackground 11 What You ll Learn To use the relationship between a radius and a tangent To use the relationship between two tangents from one point... nd Why To find the distance between the centers of two dirt bike gears, as in ample Tangent Lines heck kills You ll eed G for Help kills Handbook page 7 and Lesson 1 ind each product. 1. (p + ) p ± 6p ± 9 w ± 0w ± 100. (w + 10) m m ±. (m  ) lgebra ind the value of. Leave your answer in simplest radical form... " " ew Vocabulary tangent to a circle point of tangency inscribed in circumscribed about tangent to a circle is related to the geometric interpretation of a derivative, the fundamental concept of differential calculus. derivative measures the rate of change of a function at any point and corresponds to the slope of the tangent to the graph of the function at that point. More Math ackground: p Using the RadiusTangent Relationship Vocabulary Tip The word tangent may refer to a line, ray, or segment. In hapter, you studied the tangent ratio in right triangles. The tangents you will study here relate to circles. tangent to a circle is a line in the plane of the circle that intersects the circle in eactly one point. The point where a circle and a tangent intersect is the point of tangency. is a tangent ray and is a tangent segment. Lesson lanning and Resources Theorem 11 relates a tangent and a radius in a given circle. You will write an indirect proof for Theorem 11 in ercise 9. ee p. 660 for a list of the resources that support this lesson. oweroint ell Ringer ractice heck kills You ll eed or intervention, direct students to: kills Handbook, p. 7 The ythagorean Theorem Lesson 1: ample tra kills, Word roblems, roof ractice, h. Key oncepts Theorem 11 If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency. * ) ' You can use Theorem 11 to solve problems involving tangents to circles. 66 hapter 1 ircles 66 pecial eeds L1 Have students draw two intersecting lines and fit a quarter between the lines so that it is tangent to both lines. tudents measure the distances from the point of intersection to the points of tangency. learning style: tactile elow Level L tudents may be confused by using tangent in a new way. Try to use the phrases tangent of an angle and tangent to a circle to help reinforce the difference. learning style: verbal
2 1 ML inding ngle Measures. Teach 1 TestTaking Tip Remember that you can find the sum of the angles of a polygon with n sides using the formula (n )10. uick heck Multiple hoice ML and M are tangent to. ind the value of ince ML and M are tangent to, &L and & are right angles. LM is a quadrilateral whose angle measures have a sum of 60. m&l + m&m + m& + m& = = 60 ubstitute. The correct answer is = 60 implify. = 6 olve. 1 is tangent to. ind the value of. L 117 M Guided Instruction onnection to Language rts The term tangent is derived from the Latin verb tangere, which means to touch. Math Tip or a ray or segment to be tangent to a circle, the line containing the ray or segment must be tangent to the circle. ote that a radius is never tangent to a circle. RealWorld onnection This motorcycle and many other twowheeled vehicles have chaindrive systems like the one shown in ample. uick heck ML RealWorld onnection irt ikes dirt bike chain fits tightly around two gears. The chain and gears form a figure like the one at the right. ind the distance between the centers of the gears. 6. in.. in. Label the diagram. raw parallel to. is a rectangle. # is a right triangle with = 6. in. and = = 6.9 in. = + ythagorean Theorem = ubstitute. = 79.6 < in. implify. Use a calculator to find the square root. The distance between the centers is about 7. in. belt fits tightly around two circular pulleys, as shown at the right. ind the distance between the centers of the pulleys. about. in. dvanced Learners L fter ample, have students write an equation for the radius of a circle inscribed in an equilateral triangle with sides of length s. 1 in. 6. in.. in. in. 9. in. Theorem 1 (net page) is the converse of Theorem 11. You can use it to prove that a line or segment is tangent to a circle. You can also use it to construct a tangent to a circle (see ercise ). You will prove this theorem in ercise. learning style: verbal in. Lesson 11 Tangent Lines 66 nglish Language Learners LL ome students may confuse the terms circumscribe and inscribe. It helps students to remember that a figure that is circumscribed goes around another figure and a figure that is inscribed is in another figure. learning style: verbal ML Make sure that students remember the properties of rectangles well enough to prove that is a rectangle. sk: How do you know that & and & are right angles? n, l and l are right angles, and sameside interior angles are supplementary. How do you know that =? pposite sides of a rectangle are congruent. oweroint dditional amples 1 is tangent to at point. ind the value of. 6 belt fits tightly around two circular pulleys, as shown below. ind the distance between the centers of the pulleys. Y cm Z 1 cm 1 cm has radius. oint is outside such that = 1, and point is on such that = 1. Is tangent to at? plain. o; u ±. W 7 cm 1. cm 66
3 Guided Instruction Math Tip oint out that this lesson provides another way to define an inscribed circle: circle is inscribed in a polygon if it is tangent to each side of the polygon. Key oncepts Theorem 1 If a line in the plane of a circle is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle. * ) is tangent to. oweroint dditional amples and T are tangent to at points and T, respectively. Give a convincing argument that the diagonals of quadrilateral T are perpendicular. uick heck ML inding a Tangent Is ML tangent to at L? plain. L + LM 0 M Is kml a right triangle? ubstitute. 6 = 6 implify. y the onverse of the ythagorean Theorem, #ML is a right triangle with right &L. Therefore ML ' L, and ML is tangent to at L by Theorem 1. If L =, LM = 7, and M =, is ML tangent to at L? plain. o; ± 7 u. 7 L M T is a kite or a rhombus, so its diagonals are perpendicular. is inscribed in quadrilateral YZW. ind the perimeter of YZW. 11 ft 6 ft U T Y ft R W Z 6 ft T 7 ft 1 Using Multiple Tangents Key oncepts Theorem 1 If a circle is circumscribed about a triangle (hapter ), the triangle is inscribed in the circle. imilarly, when a circle is inscribed in a triangle, as in the diagram, the triangle is circumscribed about the circle. ach side of the triangle is tangent to the circle. The tangent segments from each verte are congruent. You will prove this theorem in ercise 0. The two segments tangent to a circle from a point outside the circle are congruent. > Resources aily otetaking Guide 11 L aily otetaking Guide 11 dapted Instruction L1 losure ind the radius of the circle inscribed in the right triangle below. in. in. uick heck 66 hapter 1 ircles ML Using Theorem 1 The diagram represents a chain drive system on a bicycle. Give a convincing argument that = G. tend and G to intersect in point H. y Theorem 1, H = H, or H + = HG + G. H y Theorem 1 again, H = HG, so by the ubtraction roperty of quality, = G. ritical Thinking Give a convincing argument that = G above if you know that and G never intersect. If and G never intersect, then G is a rectangle. G G in. 1 in. 66
4 uick heck ample 1 (page 66) amples, (pages 66, 66) ample (page 66) ML ircles Inscribed in olygons is inscribed in #. ind the perimeter of #. = = 10 cm The two segments tangent to a = = 1 cm circle from a point outside the = = cm circle are congruent. p = + + efinition of perimeter p = egment ddition ostulate = ubstitute. = 66 The perimeter is 66 cm. is inscribed in #R. #R has a perimeter of cm. ind Y. 1 cm lgebra ssume that lines that appear to be tangent are tangent. is the center of each circle. ind the value of belt fits snugly around the two circular pulleys shown.. ind the distance between the centers of the R pulleys. Round to the nearest hundredth. 1.0 in.. Give a convincing argument why the belt M lengths R and are equal. ee margin. in. in. or the pulley system shown, use the lengths given below. ind the missing length to the nearest tenth. 1 in. ercises 7 6. M = 10 cm, = cm, 7. M = in., = in., = 1 cm, M = 7 cm 1. = 0 in., M = 7 in cm 1 cm cm 1 cm Y Z 17 cm R RI or more eercises, see tra kill, Word roblem, and roof ractice. ractice and roblem olving G ractice by ample for Help Vocabulary Tip R and are common tangents.. o; ± 16 u Yes;. ± Yes; 6 ± 10. etermine whether a tangent line is shown in each diagram. plain ee left ractice ssignment Guide G nrichment Guided roblem olving Reteaching dapted ractice earson ducation, Inc. ll rights reserved. 1 ractice 1, 610, 11, 1719, 7, 9, 11, 1, 16, 0,, 0 hallenge 1 Test rep  Mied Review 96 Homework uick heck To check students understanding of key skills and concepts, go over ercises,, 1, 1, 9. ame lass ate ractice 11 Use the figures to complete ercises or figure IJKL, draw its reflection image in each line. a. ais b. yais. In the diagram, is the image of. a. ame the images of and. b. List the pairs of corresponding sides.. In the diagram, M is the image of M. a. ame the images of M and. b. List the pairs of corresponding sides.. is the image of. a. ame the images of and. b. List the pairs of corresponding sides.. or figure WYZ, draw its reflection image in each line. a. ais b. yais tate whether each transformation appears to be an isometry. plain Given points T(, ), (, ), and (0, ), draw kt and its reflection image in each line. 9. ais 10. yais 11. = 1. y = 6 6 y I J L Reflections W 6 6 Z Y y L L1 M L L L M K Lesson 11 Tangent Lines 66. tend R and until they meet at a point, H. y Thm. 11, RH H, or H ± R ± H. y 11 again, H H. Thus, R. 66
5 onnection to stronomy ercise 16 s the diagram suggests, a solar eclipse occurs when the moon blocks sunlight from reaching arth. total eclipse occurs when the moon completely blocks the sun s rays, as in the region between the moon and arth bounded by the common internal tangents to the sun and the moon. ercise tudents may benefit from a review of the properties of triangles. Tactile Learners ercise Have students manipulate a nickel, a dime, and a quarter to show how three circles can be mutually tangent. onnection to oordinate Geometry ercise 7 efore drawing a tangent segment, students must find its length,, using the ythagorean Theorem. Then they can place a compass point at (0, ), swing an arc of length until it intersects the circle, and join either point of intersection and (0, ). ercises 1 You may want students to work with partners or in small groups. uggest that they begin each proof by writing a plan. 1. a. 16a. eternal b. eternal c. internal RealWorld ample (page 66) pply Your kills onnection This diamond ring effect in a solar eclipse may be seen by a person on arth at the end of a common eternal tangent of the sun and moon. (ee diagram at right.) G nline Homework Help Visit: Hchool.com Web ode: aue101 ach polygon circumscribes a circle. ind the perimeter of the polygon in. cm 16 cm 6 cm 9 cm lgebra ssume that lines that appear to be tangent are tangent. is the center of each circle. ind the value of to the nearest tenth. in in cm 1 9 in. 7 cm 7 cm.6 cm.7 in..6 in. 1.9 in.. in. 16. olar clipse ommon tangents to two circles may be internal or eternal. If you draw a segment joining the centers of the circles, a common internal tangent will intersect the segment. common eternal tangent will not. or this crosssectional diagram of the sun, moon, and arth during a solar eclipse, use the terms above to describe the types un arth of tangents of each color. ac. ee a. red b. blue c. green above. not to scale Moon d. Which tangents show the etent blue lines; green lines on arth s surface of total eclipse? f partial eclipse? e. Reasoning In general, does every pair of circles have common tangents of both types? plain. o; eplanations may vary. ample: Two circles that have a common center have no common tangents. arth The circle at the right represents arth. The radius of arth h d is about 600 km. ind the distance d that a person can see on a r clear day from each of the following heights h above arth. r Round your answer to the nearest tenth of a kilometer m. km m 0.0 km km 11.1 km 0. and K at the right are diameters of. 7. and are tangents to. What is m&? K 1. History Leonardo da Vinci wrote, When each of G two squares touch the same circle at four points, one is double the other. ab. ee margin. a. ketch a figure that illustrates this statement. b. Writing plain why the statement is true.. Multiple hoice regular heagon is circumscribed about the ring surrounding the clock face. The diameter of the ring is 10 in. What is the perimeter of the heagon? 0.0 in..6 in.. in. 1.7 in. 666 b. nswers may vary. ample: If you draw diagonals for both squares, >are formed in the entire figure with in the small square. 666 hapter 1 ircles. 0. a and are tangent to ( at and (Given) T. # and # R (If a line is tan. to a circle, it is # to the radius.). k and k are right >. (ef.
6 .ll four are ; the two tangents to each coin from are, so by the Trans. rop., all are R or ; 1 ml is ml1. RealWorld onnection areers HV technicians often specialize in either installation or maintenance and repair of heating, ventilation, and air conditioning systems. lesson quiz, Hchool.com, Web ode: aua101 of rt. k). (Radii of a circle are.). (Refl. rop. of ) 6. k k (HL Thm.) 7. (T). ritical Thinking nickel, a dime, and a quarter are touching as shown. Tangents are drawn from point to both sides of each coin. What can you conclude about the four tangent segments? plain.. onstructions raw a circle. Label the center T. Locate a point on the circle and label it R. onstruct a tangent to T at R. ee margin. is tangent to at, and ml ind m&. 6. Let m&1 =. ind m& in terms of. What is the relationship between &1 and &? 1 7. oordinate Geometry Graph the equation + y = 9. Then draw a segment from (0, ) tangent to the circle. ind the length of the segment. ee back of book.. Maintenance Mr. Gonzales is replacing a cylindrical airconditioning duct. He estimates the radius of the duct by folding a ruler to form two 6in. tangents to the duct. The tangents form an angle. Mr. Gonzales measures the angle bisector from the verte to the duct. It is about in. long. What is the radius of the duct? about. in. 9. omplete the following indirect proof of Theorem 11. Given: Line n is tangent to at. rove: line n ' tep 1 ssume that line n is not perpendicular to. n L in. K tep If line n is not perpendicular to, some other segment L must be a. 9 to line n. y the Ruler ostulate, there is a point K on L such that K = L, so L = b. 9. #L > #KL by c. 9, so = K because d. 9. ince and K are the same distance from, both K and are on. This contradicts the given fact that line n is e. 9 to at. a. '; b. LK; c. ; d. T; e. tangent; f. false tep The assumption that line n is not perpendicular to is f. 9, so line n '. roof 0. rove Theorem 1. Given: and are tangent to at and, respectively. rove: > ee margin. 1. Given: is tangent to at.. Given: and with common > tangents and rove: > ee margin. ee rove: #G, #G margin is tangent to ( at. (Given). (Given). # (If a line is tan. to a circle, it is # to the radius.). l Lesson 11 Tangent Lines 667 G r 6 in. and l are rt. ' (ef. of #). l l (rt. ' are ) 6. (Refl. rop. of ) 7. k k (). (T). ssess & Reteach oweroint Lesson uiz and are tangent to. Use the figure for ercises 1. 0 cm 1. ind the value of. 7.. ind the perimeter of quadrilateral. cm. ind. 9 cm *HJ ) is tangent to and to. Use the figure for ercises and. H cm 9. 0 cm 1 cm J cm. ind to the nearest tenth. 0. cm. What type of special quadrilateral is HJ? plain how you know. Trapezoid; the tangent line forms right angles at vertices H and J, so H n J. ecause H u J, HJ is not a parallelogram but a trapezoid. lternative ssessment Have students use compass and straightedge to construct a circle and then construct a line through a point on the circle perpendicular to a radius of the circle. Have students repeat the construction using a different radius and then find the intersection of the two lines they constructed. fter constructing each line and finding the intersection, students should state a conclusion based on a theorem in this lesson. 667
7 Test rep Resources or additional practice with a variety of test item formats: tandardized Test rep, p. 711 TestTaking trategies, p. 706 TestTaking trategies with Transparencies hallenge roof. Write an indirect proof of Theorem 1. Given: ' at. rove: is tangent to. ee back of book.. Two circles that have one point in common are tangent circles. Given any triangle, eplain how to draw three circles that are centered at each verte of the triangle and are tangent to each other. t each verte, let the radius of a circle be the distance from the verte to either point of tangency of the incircle. Test rep. 1. ( and ( with common tangents and (Given). G G and G G (Two tan. segments from a pt. to a circle are.) G G. G 1, G 1 (iv. rop. of ) G G. G G (Trans. rop. of ). lg lg (Vert. ' are.) 6. kg M kg ( M Thm.) Multiple hoice 9. [] 9. R equivalent solution [1] correct eq. solved incorrectly hort Response Mied Review oint is the center of each circle. ssume the lines that appear tangent are tangent. What is the value of the variable? G H J J G.. 1 H.. 17 J. 9. ind the value of. how your work. ee above left. G for Help Lesson 11 Lesson  Lesson 766 hapter 1 ircles Two cubes have heights 6 in. and in. ind each ratio. 0. similarity ratio 1. ratio of surface areas. ratio of volumes : 9:16 7:6 lgebra ind the value of. Round answers to the nearest tenth The polygons are similar. (a) tate the similarity ratio and (b) find the values of the variables. a. 10:17 a. : 1 6. m b. m 1.; n a b. n 6 c b. a 1.6; b 1.7; c 66
The height of a trapezoid is the. EXAMPLE RealWorld Connection
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