# So, the measure of arc TS is 144. So, the measure of arc QTS is 248. So, the measure of arc LP is Secants, Tangents, and Angle Measures

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1 11-6 Secants, Tangents, Angle Measures Find each measure Assume that segments that appear to be tangent are tangent So, the measure of arc QTS is 48 So, the measure of arc TS is So, the measure of arc LP is STUNTS A ramp is attached to the first of several barrels that have been strapped together for a circus motorcycle stunt as shown What is the measure of the angle the ramp makes with the ground? Page 1

2 11-6 Secants, Tangents, Angle Measures So, the measure of arc LP is STUNTS A ramp is attached to the first of several barrels that have been strapped together for a circus motorcycle stunt as shown What is the measure of the angle the ramp makes with the ground? 9 Let x be the measure of the angle the ramp makes with the ground which is formed by two intersecting tangents to the circle formed by the barrel One arc has a measure of 165 The other arc is the major arc with the same endpoints, so its measure is or Therefore, the measure of the angle the ramp makes with the ground is 15 JMK HMJ form a linear pair Find each measure Assume that segments that appear to be tangent are tangent 8 11 Therefore, the measure of arc RQ is Page

3 11-6 Secants, Tangents, Angle Measures Therefore, the measure of arc RQ is By Theorem 1013, So, the measure of arc PM is By Theorem 1113, We know that Substitute Arc BD arc BCD are a minor major arc that share the same endpoints 17 SPORTS The multi-sport field shown includes a softball field a soccer field If find each measure, 15 Page 3 a b

4 We know that Substitute Simplify 11-6 Secants, Tangents, Angle Measures CCSS STRUCTURE Find each measure 17 SPORTS The multi-sport field shown includes a softball field a soccer field If find each measure, 18 By Theorem 1114, a b Substitute Simplify a By Theorem 1113, Substitute Simplify 19 b By Theorem 1114, Substitute Simplify By Theorem 1114, CCSS STRUCTURE Find each measure Substitute 18 Simplify By Theorem 1114, 0 m(arc JNM) Substitute Simplify esolutions Manual - Powered by Cognero Page 4

5 Simplify 11-6 Secants, Tangents, Angle Measures 0 m(arc JNM) By Theorem 1114, Substitute Simplify So, the measure of arc JNM is By Theorem 1114, Substitute By Theorem 1114, Substitute Simplify 4 JEWELRY In the circular necklace shown, A B are tangent points If x = 60, what is y? By Theorem 1114, Substitute Simplify By Theorem 1114, Substitute Page 5

6 Therefore, the measure of the planet s arc that is visible to the satellite is Secants, Tangents, Angle Measures ALGEBRA Find the value of x 4 JEWELRY In the circular necklace shown, A B are tangent points If x = 60, what is y? 6 By Theorem 1114, By Theorem 1114, Substitute Simplify 5 SPACE A satellite orbits above Earth s equator Find x, the measure of the planet s arc, that is visible to the satellite 7 By Theorem 1114, The measure of the visible arc is x the measure of the arc that is not visible is 360 x Use Theorem 1114 to find the value of x Therefore, the measure of the planet s arc that is visible to the satellite is ALGEBRA Find the value of x By Theorem 1114, 6 esolutions Manual - Powered by Cognero By Theorem 1114, Page 6

7 11-6 Secants, Tangents, Angle Measures 9 PHOTOGRAPHY A photographer frames a carousel in his camera shot as shown so that the lines of sight form tangents to the carousel a If the camera s viewing angle is, what is the arc measure of the carousel that appears in the shot? b If you want to capture an arc measure of in the photograph, what viewing angle should be used? 8 By Theorem 1114, 9 PHOTOGRAPHY A photographer frames a carousel in his camera shot as shown so that the lines of sight form tangents to the carousel a If the camera s viewing angle is, what is the arc measure of the carousel that appears in the shot? b If you want to capture an arc measure of in the photograph, what viewing angle should be used? a Let x be the measure of the carousel that appears in the shot By Theorem 1114, b Let x be the measure of the camera s viewing angle By Theorem 1114, a Let x be the measure of the carousel that appears in the shot By Theorem 1114, CCSS ARGUMENTS For each case of Theorem 1114, write a two-column proof 30 Case 1 Given: secants b Let x be the measure of the camera s viewing angle Page 7

8 11-6 Secants, Tangents, Angle Measures CCSS ARGUMENTS For each case of Theorem 1114, write a two-column proof 30 Case 1 Given: secants 5 (Subtraction Prop) 6 (Distributive Prop) 31 Case Given: tangent secant Statements (Reasons) 1 are secants to the circle (Given) (The, measure of an inscribed the measure of its intercepted arc) 3 Theorem) (Exterior (Substitution) 5 (Subtraction Prop) 6 (Distributive Prop) meas of an inscribed ( The the measure of its (Exterior 4 (Substitution) 5 (Subtraction Prop) 6 (Distributive Prop) 3 Case 3 Given: tangent secant is a secant to, intercepted arc) 3 Theorem) 4 31 Case Given: tangent Statements (Reasons) 1 is a tangent to the circle the circle (Given) Statements (Reasons) 1 is a tangent to the circle the circle (Given) is a secant to Statements (Reasons) 1 are tangents to the circle (Given) of an secant-tangent meas of an inscribed intercepted arc) 3 Theorem) 4 (The meas ( The, the measure of its (Exterior (Substitution) the measure of its intercepted arc) 3 Theorem) (Exterior 4 (Substitution) Page 8

9 m CAE = (Subtraction Prop) Secants, Tangents, Angle Measures (Distributive Prop) 3 Case 3 Given: tangent m(arc CA) b Prove that if CAB is obtuse, m CAB = m (arc CDA) a Proof: By Theorem 1110, So, FAE is a right with measure of 90 arc FCA is a semicircle with measure of 180 Since CAE is acute, C is in the interior of FAE, so by the Angle Arc Addition Postulates, m FAE = m FAC + m CAE m(arc FCA) = m(arc FC) + m(arc CA) By substitution, 90 = m FAC + m CAE 180 = m(arc FC) + m(arc CA) So, 90 = Statements (Reasons) 1 are tangents to the circle (Given) (The meas m(arc FC) + (arc CA) by Division Prop, m FAC + m CAE = m(arc FC) + m(arc CA) by substitution m FAC = m(arc FC) since FAC is inscribed, so substitution yields m(arc FC) + m CAE = m (arc FC) + m(arc CA) By Subtraction Prop, of an secant-tangent the measure of its intercepted arc) 3 Theorem) (Exterior 4 (Substitution) 5 (Subtraction Prop) 6 (Distributive Prop) 33 PROOF Write a paragraph proof of Theorem 1113 m CAE = m(arc CA) b Proof: Using the Angle Arc Addition Postulates, m CAB = m CAF + m FAB m(ar CDA) = m(arc CF) + m(arc FDA) Since is a diameter, FAB is a right angle with a measure of 90 arc FDA is a semicircle with a measure of 180 By substitution, m CAB = m CAF + 90 m(arc CDA) = m(arc CF) Since CAF is inscribed, m CAF = m(arc CF) by substitution, m CAB = m(arc CF) + 90 Using the Division Subtraction Properties on the Arc Addition equation yields m(arc CDA) m(arc CF) = 90 By substituting for 90, m CAB = CF) + m(arc CDA) m(arc m(arc CF) Then by subtraction, m CAB = m(arc CDA) a Given: is a tangent of is a secant of CAE is acute m CAE = m(arc CA) b Prove that if CAB is obtuse, m CAB = m (arc CDA) esolutions Manual - Powered by Cognero a Proof: By Theorem 1110, So, FAE is a right with measure of 90 arc FCA is a 34 WALLPAPER The design shown is an example of optical wallpaper is a diameter of If m A = 6 = 67, what is Refer to the image on page 766 First, find the measure of arc BD Page 9

10 11-6 Secants, Tangents, Angle Measures a Redraw the circle with points A, B, C in the same place but place point D closer to C each time Then draw chords 34 WALLPAPER The design shown is an example of optical wallpaper is a diameter of If m A = 6 = 67, what is Refer to the image on page 766 b First, find the measure of arc BD Since is a diameter, arc BDE is a semicircle has a measure of 180 c As the measure of gets closer to 0, the measure of x approaches half of becomes an inscribed angle 35 MULTIPLE REPRESENTATIONS In this problem, you will explore the relationship between Theorems a GEOMETRIC Copy the figure shown Then draw three successive figures in which the position of point D moves closer to point C, but points A, B, C remain fixed b TABULAR Estimate the measure of for each successive circle, recording the measures of in a table Then calculate record the value of x for each circle d 36 WRITING IN MATH Explain how to find the measure of an angle formed by a secant a tangent that intersect outside a circle Find the difference of the two intercepted arcs divide by 37 CHALLENGE The circles below are concentric What is x? c VERBAL Describe the relationship between the value of x as approaches zero What type of angle does AEB become when d ANALYTICAL Write an algebraic proof to show the relationship between Theorems described in part c Use the arcs intercepted on the smaller circle to find m A a Redraw the circle with points A, B, C in the same place but place point D closer to C each time Then draw chords b Use the arcs intercepted on the larger circle Page 10 m A to find the value of x

11 tangent that intersect outside a circle the difference of the intercepted arcs 11-6Find Secants, Tangents, two Angle Measures divide by 37 CHALLENGE The circles below are concentric What is x? because the triangle is isosceles Since BAC BCA are inscribed angles, by Theorem 116, m (arc AB) = m BCA m(arc BC) = m BAC So, m(arc AB) = m(arc BC) 39 CCSS ARGUMENTS In the figure, is a diameter is a tangent a Describe the range of possible values for m G Explain b If m G = 34, find the measures of minor arcs HJ KH Explain a ; at G, then Use the arcs intercepted on the smaller circle to find m A for all values except when b Because a diameter is involved the intercepted arcs measure (180 x) leads to x degrees Hence solving the answer Use the arcs intercepted on the larger circle m A to find the value of x 40 OPEN ENDED Draw a circle two tangents that intersect outside the circle Use a protractor to measure the angle that is formed Find the measures of the minor major arcs formed Explain your reasoning Sample answer: 38 REASONS Isosceles is inscribed in What can you conclude about Explain By Theorem 1113, So, 50 = Therefore, x (minor arc) = 130, y (major arc) = or 30 Sample answer: m BAC = m BCA because the triangle is isosceles Since BAC BCA are inscribed angles, by Theorem 116, m (arc AB) = m BCA m(arc BC) = m BAC 41 WRITING IN MATH A circle is inscribed within If m P = 50 m Q = 60, describe how to find the measures of the three minor arcs formed by the points of tangency So, m(arc AB) = m(arc BC) 39 CCSS ARGUMENTS In the figure, is a diameter is a tangent a Describe the range of possible values for m G Explain Page 11

12 By Theorem 1113, So, 50 = Therefore, x (minor arc) = 130, 11-6 Secants, Tangents, Angle Measures y (major arc) = or WRITING IN MATH A circle is inscribed within If m P = 50 m Q = 60, describe how to find the measures of the three minor arcs formed by the points of tangency Therefore, the measures of the three minor arcs are 130, 10, What is the value of x if A 3 B 31 C 64 D 18 Sample answer: Use Theorem 1114 to find each minor arc By Theorem 1114, Substitute Simplify The sum of the angles of a triangle is 180, so m R = 180 ( ) or 70 Therefore, the measures of the three minor arcs are 130, 10, 110 So, the correct choice is C 4 What is the value of x if 43 ALGEBRA Points A( 4, 8) B(6, ) are both on circle C, is a diameter What are the coordinates of C? F (, 10) G (10, 6) H (5, 3) J (1, 5) A 3 B 31 C 64 D 18 Here, C is the midpoint of Use the midpoint formula, By Theorem 1114,, to find the coordinates of C Substitute Simplify So, the correct choice is J Page 1 44 GRIDDED RESPONSE If m AED = 95

13 Substitute 11-6 Secants, Tangents, Angle Measures So, the correct choice is C 43 ALGEBRA Points A( 4, 8) B(6, ) are both on circle C, is a diameter What are the coordinates of C? F (, 10) G (10, 6) H (5, 3) J (1, 5), Since 45 SAT/ACT If the circumference of the circle below is 16π units, what is the total area of the shaded regions? A 64π units B 3π units Here, C is the midpoint of Use the midpoint formula, C 1π units D 8π units, to find the coordinates of D π units C First, use the circumference to find the radius of the circle So, the correct choice is J 44 GRIDDED RESPONSE If m AED = 95 what is m BAC? The shaded regions of the circle comprise half of the circle, so its area is half the area of the circle Since We know that vertical angles are congruent So, Since, We know that the sum of the measures of all interior angles of a triangle is 180 The area of the shaded regions is 3π So, the correct choice is B Find x Assume that segments that appear to be tangent are tangent Substitute 46, Since 45 SAT/ACT If the circumference of the circle below is 16π units, what is the total area of the shaded regions? By Theorem 1110, So, triangle Use the Pythagorean Theorem is a right Substitute A 64π units B 3π units Page 13

14 area oftangents, the shaded regions is 3π 11-6The Secants, Angle Measures So, the correct choice is B Find x Assume that segments that appear to be tangent are tangent 49 PROOF Write a two-column proof Given: is a semicircle 46 By Theorem 1110, So, triangle Use the Pythagorean Theorem is a right Given: Substitute is a semicircle 47 If two segments from the same exterior point are tangent to a circle, then they are congruent Proof: Statements (Reasons) 1 MHT is a semicircle; (Given) is a right angle (If an inscribed angle intercepts a semicircle, the angle is a right angle) 3 is a right angle (Def of lines) 4 (All rt angles are ) 5 (Reflexive Prop) 6 (AA Sim) 7 48 Use Theorem 1110 the Pythagorean Theorem x (Def of COORDINATE GEOMETRY Find the measure of each angle to the nearest tenth of a degree by using the Distance Formula an inverse trigonometric ratio 50 C in triangle BCD with vertices B( 1, 5), C( 6, 5), D( 1, ) Use the Distance Formula to find the length of each side 49 PROOF Write a two-column proof Given: is a semicircle Page 14 Since triangle, triangle BCD is a right

15 4 (All rt angles are ) 5 (Reflexive Prop) 6 (AA Sim) 11-67Secants, Tangents, (Def of Angle Measures COORDINATE GEOMETRY Find the measure of each angle to the nearest tenth of a degree by using the Distance Formula an inverse trigonometric ratio 50 C in triangle BCD with vertices B( 1, 5), C( 6, 5), D( 1, ) If Use a calculator 51 X in right triangle XYZ with vertices X(, ), Y(, ), Z(7, ) Use the Distance Formula to find the length of each side Use the Distance Formula to find the length of each side In right triangle XYZ, ZX is the length of the hypotenuse YZ is the length of the leg opposite Write an equation using the sine ratio Since, triangle BCD is a right triangle So, CD is the length of the hypotenuse BC is the length of the leg adjacent to Write an equation using the cosine ratio If Use a calculator If Use a calculator Solve each equation 5 x + 13x = X in right triangle XYZ with vertices X(, ), Y(, ), Z(7, ) Use the Distance Formula to find the length of each side Therefore, the solution is 9, 4 53 x 6x = 9 In right triangle XYZ, ZX is the length of the hypotenuse YZ is the length of the leg opposite Therefore, the solution is 3 Write an equation using the sine ratio 54 3x + 15x = 0 If Use a calculator Therefore, the solution is 5, 0 Page 15

16 11-6 Secants, Tangents, Angle Measures Therefore, the solution is x + 15x = 0 Therefore, the solution is 5, = x + 3x Therefore, the solution is 7, 4 56 x + 1x + 36 = 0 Therefore, the solution is 6 Page 16

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