10-7 Special Segments in a Circle. Find x. Assume that segments that appear to be tangent are tangent. 1. SOLUTION: ANSWER: 2 SOLUTION: ANSWER:
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1 Find x. Assume that segments that appear to be tangent are tangent esolutions Manual - Powered by Cognero Page 1
2 5. SCIENCE A piece of broken pottery found at an archaeological site is shown. lies on a diameter of the circle. What was the circumference of the original pottery? Round to the nearest hundredth. 7. Let be the diameter of the circle. Then and are intersecting chords. Find the length of the diameter by first finding the measure of ST. 5 The diameter is equal to QS + ST or cm Find x to the nearest tenth. Assume that segments that appear to be tangent are tangent esolutions Manual - Powered by Cognero Page 2
3 Solve for x by using the Quadratic Formula. Solve for x using the Quadratic Formula. Since lengths cannot be negative, the value of x is about Since lengths cannot be negative, the value of x is about esolutions Manual - Powered by Cognero Page 3
4 14. BRIDGES What is the diameter of the circle containing the arc of the Sydney Harbour Bridge? Round to the nearest tenth. Refer to the photo on Page CAKES Sierra is serving cake at a party. If the dimensions of the remaining cake are shown below, what was the original diameter of the cake? Since the height is perpendicular and bisects the chord, it must be part of the diameter. To find the diameter of the circle, first use Theorem to find the value of x. Add the two segments to find the diameter. Therefore, the diameter of the circle is or about meters m Since one of the perpendicular segments contains the center of the cake, it will bisect the chord. To find the diameter of the cake, first use Theorem to find x. Add the two segments to find the diameter. The diameter of the circle is or 13 inches. 13 in. esolutions Manual - Powered by Cognero Page 4
5 CCSS STRUCTURE Find each variable to the nearest tenth. Assume that segments that appear to be tangent are tangent Solve for x using the Quadratic Formula. First, find the length, a, of the tangent segment to both circles using the intersecting secant and tangent segments to the smaller circle. Since lengths cannot be negative, the value of x is 6. 6 Find x using the intersecting secant and tangent segments to the larger circle Solve for x using the Quadratic Formula. Since lengths cannot be negative, the value of x is about esolutions Manual - Powered by Cognero Page 5
6 19. Use the intersecting tangent and secant segments to find a. 20. Use the intersecting tangent and secant segments to find q. Next, use the intersecting chords to find b. Since lengths cannot be negative, the value of q is 9. Use the intersecting secant segments to find r. a = 15; b 11.3 Solve for r using the Quadratic Formula. Since lengths cannot be negative, the value of r is about 1.8. Therefore, the values of the variables are q = 9 and r 1.8. q = 9; r 1.8 esolutions Manual - Powered by Cognero Page 6
7 21. Use the intersecting secant segments to find c. 22. INDIRECT MEASUREMENT Gwendolyn is standing 16 feet from a giant sequoia tree and Chet is standing next to the tree, as shown. The distance between Gwendolyn and Chet is 27 feet. Draw a diagram of this situation, and then find the diameter of the tree. Use the intersecting secant and tangent segments to find d. Therefore, the values of the variables are c 22.8 and d c 22.8; d 16.9 Let d be the diameter of the tree. Use the intersecting tangent and secant segments to find d. So, the diameter is about 29.6 ft ft esolutions Manual - Powered by Cognero Page 7
8 PROOF Prove each theorem. 23. two-column proof of Theorem Given: and intersect at B. Prove: 24. paragraph proof of Theorem Given: Secants and Prove: Proof: Statements (Reasons) 1. and intersect at B. (Given) 2., (Inscribed that intercept the same arc are 3. (AA Similarity) 4. (Definition of similar 5. (Cross products) Proof: Statements (Reasons) 1. and intersect at B. (Given) 2., (Inscribed that intercept the same arc are 3. (AA Similarity) 4. (Definition of similar 5. (Cross products) Proof: and are secant segments. By the Reflexive Property,. Inscribed angles that intercept the same arc are congruent. So,. By AA Similarity, By the definition of similar triangles, are equal, Proof: Since the cross products of a proportion and are secant segments. By the Reflexive Property,. Inscribed angles that intercept the same arc are congruent. So,. By AA Similarity, By the definition of similar triangles, are equal, Since the cross products of a proportion esolutions Manual - Powered by Cognero Page 8
9 25. two-column proof of Theorem Given: tangent secant Prove: 26. CRITIQUE Tiffany and Jun are finding the value of x in the figure at the right. Tiffany wrote 3(5) = 2x, and Jun wrote 3(8) = 2(2 + x). Is either of them correct? Explain your reasoning. Proof: Statements (Reasons) 1. tangent and secant (Given) 2. (The measure of an inscribed equals half the measure of its intercept arc.) 3. (The measure of an formed by a secant and a tangent = half the measure of its intercepted arc.) 4. (Substitution) 5. (Definition of 6. (Reflexive Property) 7. (AA Similarity) Jun; the segments intersect outside of the circle, so the correct equation involves the products of the measures of a secant and its external secant segment. Jun; the segments intersect outside of the circle, so the correct equation involves the products of the measures of a secant and its external secant segment. 8. (Definition of 9. (Cross products) Proof: Statements (Reasons) 1. tangent and secant (Given) 2. (The measure of an inscribed equals half the measure of its intercept arc.) 3. (The measure of an formed by a secant and a tangent = half the measure of its intercepted arc.) 4. (Substitution) 5. (Definition of 6. (Reflexive Property) 7. (AA Similarity) 8. (Definition of 9. (Cross products) esolutions Manual - Powered by Cognero Page 9
10 27. WRITING IN MATH Compare and contrast the methods for finding measures of segments when two secants intersect in the exterior of a circle and when a secant and a tangent intersect in the exterior of a circle. Sample answer: When two secants intersect in the exterior of a circle, the product of the measures of one secant segment and its external segment is equal to the product of the measures of the other secant segment and its external segment. When a secant and a tangent intersect at an exterior point, the product of the measures of the secant segment and its external segment is equal to the square of the measure of the tangent segment, because for the tangent the measures of the external segment and the whole segment are the same. Sample answer: When two secants intersect in the exterior of a circle, the product of the measures of one secant segment and its external segment is equal to the product of the measures of the other secant segment and its external segment. When a secant and a tangent intersect at an exterior point, the product of the measures of the secant segment and its external segment is equal to the square of the measure of the tangent segment, because for the tangent the measures of the external segment and the whole segment are the same. 28. CHALLENGE In the figure, a line tangent to circle M and a secant line intersect at R. Find a. Show the steps that you used. Use the intersecting tangent and secant segments to find the value of a. esolutions Manual - Powered by Cognero Page 10
11 29. REASONING When two chords intersect at the center of a circle, are the measures of the intercepting arcs sometimes, always, or never equal to each other? Since the chords intersect at the center, the measure of each intercepted arc is equal to the measure of its related central angle. Since vertical angles are congruent, the opposite arcs will be congruent. All four arcs will only be congruent when all four central angles are congruent. This will only happen when the chords are perpendicular and the angles are right angles Therefore, the measures of the intercepted arcs are sometimes equal to each other. Sometimes; they are equal when the chords are perpendicular. 30. OPEN ENDED Investigate Theorem by drawing and labeling a circle that has a secant and a tangent intersecting outside the circle. Measure and label the two parts of the secant segment to the nearest tenth of a centimeter. Use an equation to find the measure of the tangent segment. Verify your answer by measuring the segment. Sample answer: Therefore, the measure of the tangent segment is about 2.4 centimeters. Sample answer: x 2.4 cm 31. WRITING IN MATH Describe the relationship among segments in a circle when two secants intersect inside a circle. Sample answer: The product of the parts on one intersecting chord equals the product of the parts of the other chord. Sample answer: The product of the parts on one intersecting chord equals the product of the parts of the other chord. esolutions Manual - Powered by Cognero Page 11
12 32. is tangent to the circle, and R and S are points on the circle. What is the value of x to the nearest tenth? A 7.6 B 6.4 C 5.7 D ALGEBRA A department store has all of its jewelry discounted 40%. It is having a sale that says you receive an additional 20% off the already discounted price. How much will you pay for a ring with an original price of $200? F $80 G $96 H $120 J $140 Discount price = = 120 Solve for x using the Quadratic Formula. Discount price = = 96 So, the correct choice is G. G Since length can not be negative, the value of x is 5.7. Therefore, the correct answer is C. C esolutions Manual - Powered by Cognero Page 12
13 34. EXTENDED RESPONSE The degree measures of minor arc AC and major arc ADC are x and y, respectively. a. If m ABC = 70, write two equations relating x and y. b. Find x and y. 36. WEAVING Once yarn is woven from wool fibers, it is often dyed and then threaded along a path of pulleys to dry. One set of pulleys is shown. Note that the yarn appears to intersect itself at C, but in reality it does not. Use the information from the diagram to find a. Since the sum of the arcs of a circle is 360, x + y = 360, and by Theorem 10.14, y x = 140. b. First, use intersecting secants to find m GCD. By vertical angles, m BCH = m GCD. So, m BCH = 39. Since major arc BAH has the same endpoints as minor arc BH, m(arc BAH) = 360 m(arc BH). Therefore, x = 110 and y = 250. a. x + y = 360 and y x = 140 b. x = 110, y = SAT/ACT During the first two weeks of summer vacation, Antonia earned $100 per week. During the next six weeks, she earned $150 per week. What was her average weekly pay? A $50 B $ C $125 D $135 E $ Therefore, the measure of arc BH is Simplify. So, the correct choice is E. E esolutions Manual - Powered by Cognero Page 13
14 Copy the figure shown and draw the common tangents. If no common tangent exists, state no common tangent. 37. Four common tangents can be drawn to these two circles. 39. Three common tangents can be drawn to these two circles. 38. Every tangent drawn to the small circle will intersect the larger circle in two points. Every tangent drawn to the large circle will not intersect the small circle at any point. Since a tangent must intersect the circle at exactly one point, no common tangent exists for these two circles. no common tangent 40. Two common tangents can be drawn to these two circles. esolutions Manual - Powered by Cognero Page 14
15 COORDINATE GEOMETRY Graph each figure and its image along the given vector. 41. with vertices K(5, 2), L( 3, 1), and M(0, 5); 42. quadrilateral PQRS with vertices P(1, 4), Q( 1, 4), R ( 2, 4), and S(2, 4); The coordinates of the vertices of the translated triangle are K (2, 6), L ( 6, 5), and M ( 3, 1). The coordinates of the vertices of the translated quadrilateral are P ( 4, 7), Q ( 6, 7), R ( 7, 1), and S ( 3, 1). esolutions Manual - Powered by Cognero Page 15
16 43. with vertices E(0, 4), F( 4, 4), and G(0, 2); 45. m: 2, (0, 8) To find the slope-intercept form of the equation of this line use the point-slope form of a line, y y 1 = m (x x 1 ) where m is the slope and (x 1, y 1 ) is a point on the line. Here, m = 2 and (x 1, y 1 ) = (0, 8). The coordinates of the vertices of the translated triangle are E (2, 5), F ( 2, 5), and G (2, 1). y = 2x To find the slope-intercept form of the equation of this line use the point-slope form of a line, y y 1 = m (x x 1 ) where m is the slope and (x 1, y 1 ) is a point on the line. Here, and (x 1, y 1 ) = (0, 6). Write an equation in slope-intercept form of the line having the given slope and y-intercept. 44. m: 3, y-intercept: 4 The slope-intercept form of a line of slope m and y- intercept b is given by y = mx + b. For this line, substitute m = 3 and b = 4. So, the equation of the line is y = 3x 4. y = 3x 4 esolutions Manual - Powered by Cognero Page 16
17 47. The slope-intercept form of a line of slope m and y- intercept b is given by y = mx + b.for this line, substitute and b =. So, the equation of the line is. 48. m: 1, b: 3 The slope-intercept form of a line of slope m and y- intercept b is given by y = mx + b. For this line, substitute and b = 3. So, the equation of the line is y = 1x 3 or y = x 3. y = x The slope-intercept form of a line of slope m and y- intercept b is given by y = mx + b. For this line, substitute and b = 1. So, the equation of the line is. esolutions Manual - Powered by Cognero Page 17
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