10-7 Special Segments in a Circle. Find x. Assume that segments that appear to be tangent are tangent. 1. SOLUTION: 2. SOLUTION: 3.
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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 4. 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. 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. The diameter is equal to QS + ST or. esolutions Manual - Powered by Cognero Page 2
3 Find x to the nearest tenth. Assume that segments that appear to be tangent are tangent esolutions Manual - Powered by Cognero Page 3
4 esolutions Manual - Powered by Cognero Page 4
5 11. Solve for x by using the Quadratic Formula. Since lengths cannot be negative, the value of x is about esolutions Manual - Powered by Cognero Page 5
6 13. Solve for x using the Quadratic Formula. Since lengths cannot be negative, the value of x is about 7.4. esolutions Manual - Powered by Cognero Page 6
7 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 740. 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. esolutions Manual - Powered by Cognero Page 7
8 15. 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 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. esolutions Manual - Powered by Cognero Page 8
9 Find each variable to the nearest tenth. Assume that segments that appear to be tangent are tangent. 16. Solve for x using the Quadratic Formula. Since lengths cannot be negative, the value of x is Solve for x using the Quadratic Formula. Since lengths cannot be negative, the value of x is about 7.1. esolutions Manual - Powered by Cognero Page 9
10 18. First, find the length, a, of the tangent segment to both circles using the intersecting secant and tangent segments to the smaller circle. Find x using the intersecting secant and tangent segments to the larger circle. 19. Use the intersecting tangent and secant segments to find a. Next, use the intersecting chords to find b. esolutions Manual - Powered by Cognero Page 10
11 20. Use the intersecting tangent and secant segments to find q. Since lengths cannot be negative, the value of q is 9. Use the intersecting secant segments to find r. 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. esolutions Manual - Powered by Cognero Page 11
12 21. Use the intersecting secant segments to find c. Use the intersecting secant and tangent segments to find d. Therefore, the values of the variables are c 22.8 and d 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. 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. esolutions Manual - Powered by Cognero Page 12
13 PROOF Prove each theorem. 23. two-column proof of Theorem Given: and intersect at B. 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) 24. paragraph proof of Theorem Given: Secants Prove: and 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, Since the cross products of a proportion are equal, esolutions Manual - Powered by Cognero Page 13
14 25. two-column proof of Theorem Given: tangent secant Prove: 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) 26. ERROR ANALYSIS 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. 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. 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. esolutions Manual - Powered by Cognero Page 14
15 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. 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. esolutions Manual - Powered by Cognero Page 15
16 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. 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. esolutions Manual - Powered by Cognero Page 16
17 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 4.8 Solve for x using the Quadratic Formula. Since length can not be negative, the value of x is 5.7. Therefore, the correct answer is C. esolutions Manual - Powered by Cognero Page 17
18 33. 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 Discount price = = 96 So, the correct choice is G. 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. a. Since the sum of the arcs of a circle is 360, x + y = 360, and by Theorem 10.14, y x = 140. b. Therefore, x = 110 and y = 250. esolutions Manual - Powered by Cognero Page 18
19 35. 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 $ Simplify. So, the correct choice is E. 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 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, the measure of arc BH is 141. esolutions Manual - Powered by Cognero Page 19
20 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. 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. 39. Three common tangents can be drawn to these two circles. esolutions Manual - Powered by Cognero Page 20
21 40. Two common tangents can be drawn to these two circles. 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); The coordinates of the vertices of the translated triangle are K (2, 6), L ( 6, 5), and M ( 3, 1). esolutions Manual - Powered by Cognero Page 21
22 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 quadrilateral are P ( 4, 7), Q ( 6, 7), R ( 7, 1), and S ( 3, 1). 43. with vertices E(0, 4), F( 4, 4), and G(0, 2); The coordinates of the vertices of the translated triangle are E (2, 5), F ( 2, 5), and G (2, 1). 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. esolutions Manual - Powered by Cognero Page 22
23 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). 46. 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). 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. esolutions Manual - Powered by Cognero Page 23
24 49. 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 24
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