i8o Chapter 12, Lesson 1 Chapter 12, Lesson 1 •14. They are perpendicular. 15. The longest chord is always a diameter; so Set I (pages 486-487) its length doesn't depend on the location of According to Matthys Levy and Mario Salvador! the point. The length of the shortest chord in their book titled \Nhy Buildings Fall Down depends on how far away from the center (Norton, 1992), the Romans built semicircular the point is; the greater the distance, the arches of the type illustrated in exercises 1 through shorter the chord. 5 that spanned as much as 100 feet. These arches Theorem 56. were used in the bridges of 50,000 miles of roads that the Romans built all the way from Baghdad 16. Perpendicular lines form right angles. to London! • 17. Two points determine a line. The conclusions of exercises 13 through 15 are supported by the fact that a diameter of a circle is •18. All radii of a circle are equal. its longest chord and by symmetry considerations. 19. Reflexive. As the second chord rotates away from the diameter in either direction about the point of 20. HL. intersection, it becomes progressively shorter. 21. Corresponding parts of congruent triangles are equal. Roman Arch. •22. If a line divides a line segment into two 1. Radii. equal parts, it bisects the segment. •2. Chords. Theorem 57. •3. A diameter. 23. If a line bisects a line segment, it divides the 4. Isosceles. Each has two equal sides because segment into two equal parts. the radii of a circle are equal. 24. Two points determine a line. 5. They are congruerit by SSS. 25. All radii of a circle are equal. Old Figure. • 26. In a plane, two points each equidistant from •6. Five. the endpoints of a line segment determine the perpendicular bisector of the line 7. Oi\e. segment. 8. Four. Theorem 58. •9. Six. •27. If a line through the center of a circle bisects Butterfly Spots. a chord that is not a diameter, it is also perpendicular to the chord. • 10. Concentric. • 28. Through a point on a line, there is exactly 11. It means that a set of circles lie in the same one line perpendicular to the line. plane and have the same center. 29. CD contains O. 12. (Student answer.) (The spots look like big eyes; so they might scare some predators Set II (pages 488-490) away.) Chord Problem. The SAT problem of exercises 30 and 31 is something of an oddity in that it is a rather well 13. known "trick question." Martin Gardner included it in his "Mathematical Games" column in the November 1957 issue of Scientific American- many years before it appeared on the SAT! The CBS eye, a creation of designer William Golden, first appeared on television in 1951. Hal Chapter 12, Lesson 1 181 Morgan observes in his bool< titled Symbols of CBS Eye. America (Viking, 1986) that, "because it is 32. Example answer: essentially a picture of an eye, it has proved a challenge to the police efforts of the CBS trademark attorneys. As one company executive admitted, 'Bill Golden, after all, didn't invent the eye. God thought of it first.'" The story of the use of Jupiter to determine the speed of light is told by William Dunham in The Mathematical Universe (Wiley, 1994). After the invention of the telescope in the seventeenth century and the discovery of the moons of Jupiter, astronomers were puzzled by the delay in the eclipses of the moon lo when the earth and Jupiter were farthest apart. Ole Roemer reasoned that the time discrepancy was due to the extra 33. The perpendicular bisector of a chord of a distance that the light from Jupiter and its moon circle contains the center of the circle. had to travel to get to the earth. Surprisingly, it was not the speed of light that especially interested Red-Spot Puzzle. Roemer but rather the conclusion that light does •34. An isosceles right triangle. not travel instantaneously from one place to another, something that had not been known previously. •35. About 46 mm. (-^=45.96.) V2 The definition of the roundness of a grain of sand was devised by Haakon Wadell in about 36. Very difficult because the disks are barely 1940. He defined it to be (for cross sections of the big enough to cover the spot. grain) the "ratio of average radius of curvature Speed of Light. of the corners to radius of maximum inscribed circle." From this definition, it follows that 37. 186,000,000 mi. (2 x 93,000,000.) roundness can vary from o to 1, with perfect •38. 1,000 seconds. (16 x 60 + 40.) roundness being 1. Exercises 43 through 47 are based on a simple 186 000 000 theorem discovered in 1916 by the American 39. 186,000 miles per second. ( 77;-^ •) geometer Roger Johnson: "If three circles having 1, uuu the same radius pass through a point, the circle Roundness. through their other three points of intersection also has the same radius." The appearance of the •40. |, or about 0.37. [^||^.] cube in the explanation and the role played by two of its opposite vertices is quite unexpected. 41. ^ = l^,orabout0.37.[i±^.] For more information on this elegant theorem, 54 27 3(18) see chapter 10 of George Polya's Mathematical Discovery, vol. 2 (Wiley, 1965), and the fourth of 42. No. The figure would be larger but still the "Four Minor Gems from Geometry" in Ross have the same shape. Honsberger's Mathematical Cems II (Mathematical Association of America, 1976). SAT Problem. 30. 10. 31. The diagonals of a rectangle are equal. The other diagonal of the rectangle is a radius of the circle. i82 Chapter 12, Lesson 1 Three Circles. Set Hi (page490) 43. When this puzzle appeared in Martin Gardner's "Mathematical Games" column in Scientific American, William B. Friedman suggested tying the rope to two places high on the tree at B so that the person could walk along the lower part while holding onto the upper part and not get wet. Island Puzzle. Tie one end of the rope to the tree at B. Holding the other end of the rope, walk around the lake so that the rope wraps around the tree at A. Pull the rope tight and tie it at the other end to the tree at B. Hold onto the rope to get to the island. 44. They are each 1 unit. •45. They are rhombuses because all of their Chapter 12, Lesson 2 sides are equal. Set I (pages 493-494) 46. They are equal. The opposite sides of a parallelogram are equal and the other Exercises 15 through 18 reveal how the tangent segments all equal 1 unit. ratio got its name. As Howard Eves remarks in An Introduction to the History of Mathematics 47. The circle would have its center at O (Saunders, 1990), "the meanings of the present because OA = OB = OC = 1. names of the trigonometric functions, with the Steam Engine. exception of sine, are clear from their geometrical interpretations when the angle is placed at the 48. center of a circle of unit radius The functions A tangent, cotangent, secant, and cosecant have been known by various other names, but these 16 particular names appeared as late as the end of c the 16th century." 12 E Band Saw. -V50 1. If a line is tangent to a drde, it is 14 perpendicular to the radius drawn to the B •••A \ point of contact. 14 n 1 \ •2. In a plane, two lines perpendicular to a 14 14 ID } third line are parallel to each other. F 3. The opposite sides of a parallelogram are equal. Concentric Circles. 49. EF = 50 in. [EP + PF = 8 + 3(14).] •4. OBI AC. 50. CD = 50in.(CD = EF.) •5. If a line through the center of a circle bisects a chord that is not a diameter, it is also 51. CO = 48iiMCO2 + 142 = 502;so perpendicular to the chord. CO = VSO^-M^ = V2,304 = 48.) 6. AC is tangent to the smaller circle. 52. CE = 12 in. (CE = CO - EO = 48 - 36.) 7. If a line is perpendicular to a radius at its 53. AC = 16 in. (AC = AB - CB = 50 - 34.) outer endpoint, it is tangent to the circle. Chapter 12, Lesson 2 183 Grid Exercise. Tangents to Two Circles. 8. 5. -I- .10. -2. 4 11. -I 4 12. Lines Z and m are perpendicular to line AB. 13. Lines / and m are parallel. 14. The slopes of I and m are equal; the slope of AB is the opposite of the reciprocal of their slope. 26. Example figure: Double Meaning. • 15. AB is tangent to circle O. • 16. If a line is perpendicular to a radius at its outer endpoint, it is tangent to the circle. 17. The tangent of an acute angle of a right triangle is the ratio of the length of the 27. Example figure: opposite leg to the length of the adjacent leg. •18. ^,orAB. Railroad Curve. 28. Yes. No lines can be drawn tangent to both circles if one circle is inside the other and 1 ^ the circles have no point in common. Set 11 (pages 494-496) ^ / Euclid's Book 111, Proposition 17 is the following construction: "From a given point to draw a 0 straight line touching a given circle." A construction based on the fact that an angle inscribed in a semicircle is a right angle is introduced in Lesson 20.