EXERCISES. THEOREMS. 1. If AB be a side of an equilateral triangle inscribed in a circle, and AD a side of the inscribed square: prove that three times the square on AD is equal to twice the square on AB. 2. Show that the sum of the perpendiculars from any point inside a regular hexagon to the six sides is equal to three times the diameter of the inscribed circle. 3. The area of the regular inscribed hexagon is twice the area of the inscribed equilateral triangle. 4. The area of the regular inscribed hexagon is threefourths of that of the regular circumscribed hexagon. 5. The area of the regular inscribed hexagon is a mean proportional between the areas of the inscribed and circumscribed equilateral triangles. 6. If the perpendicular from A to the side BC of the equilateral triangle ABC meet BC in D, and the inscribed circle in G; prove that GD = 2AG. See figure of (269). 7. If three circles touch each other externally, and a triangle be formed by joining their centres, and another triangle by joining their points of contact, the inscribed circle of the former triangle will be the circumscribed circle of the latter. 8. If ABCD be a square described about a given circle, and P any point on the circumference of the circle; prove that the sum of the squares on PA, PB, PC, PD, is three times the square on the diameter of the circle. Use (333). 9. ABC is a triangle having each of the angles B, C double the angle A; the bisectors of the angles B, C meet AC and the circle circumscribing the triangle ABC respectively in D, E: prove that ADBE is a rhombus. 10. ABCDE is a regular pentagon inscribed in a circle, P is the middle point of the arc AB: show that the difference of the straight lines AP, CP is equal to the radius of the circle squares 11. In the last exercise prove that the sum of the on PC and BC is equal to four times the square on the radius. 12. ABCDE is a regular pentagon, BE is drawn cutting AC, AD in F, G respectively: show that the sum of AB and AE is equal to the sum of BE and FG. Describe a about the pentagon, etc. 13. In a circle a regular pentagon and a regular decagon are inscribed; the middle points of the adjacent sides of the pentagon are joined: prove that the sides of the pentagon so formed are equal to the radius of the circle inscribed in the decagon. 14. Every equilateral polygon inscribed in a circle is equiangular. 15. Every equilateral polygon circumscribed about a circle is equiangular, if the number of sides be odd. 16. Every equiangular polygon inscribed in a circle is equilateral, if the number of sides be odd. 17. Every equiangular polygon circumscribed about a circle is equilateral. 18. The figure formed by the five diagonals of a regular pentagon is another regular pentagon. 19. If the alternate sides of a regular pentagon be produced to meet, the five points of meeting form another regular pentagon. Let a denote a side of a regular polygon inscribed in a circle whose radius is R; then: 25. The side of the regular inscribed pentagon is equal to the hypotenuse of a right triangle whose sides are the radius and the side of the regular inscribed decagon. NUMERICAL EXERCISES. Ans. 3.535 ft. 26. The diameter of a circle is 5 feet: find the side of the inscribed square. 27. The apothem of a regular hexagon is 2: find the area of the circumscribing circle. Ans. 5 π. 28. There are two gardens; one is a square, and the other is a circle; and they each contain an acre: how much further is it around one than the other? Ans. 17.268 yards. 29. There are two circles; the diameter of the first is 18 inches, and the area of the second is 23 times that of the first: what is the diameter of the second? Ans. 30 inches. 30. The circumference of a circle is 78.54 inches: find (1) its diameter, and (2) its area. Ans. (1) 25 ins.; (2) 490.875 sq. ins. 31. A circle and a square have each a perimeter of 120 feet: which contains the greater area, and how much? Ans. The circle, 245.95 sq. ft. 32. What is the width of a ring between two concentric circumferences whose lengths are 160 feet and 80 feet? Ans. 12.732 feet. 33. Find the side of a square equivalent to a circle whose radius is 40 feet. 34. The radius of a circle is 15 feet: find the radius of a circle just three times as large. 35. The area of a square is 225 square feet: find the area of the inscribed circle. 36. The length of an arc of 180° is πR, where R is the radius of the circle (436): find the length of the arc of 25° 45′ in the circle whose radius is 9 inches. Ans. 4 ins. 37. Find the number of degrees in the arc whose length is equal to the radius of the circle. Ans. 57° 17′ 44′′.8. 38. Find the number of degrees in the arc whose length is 18 inches, the radius being 5 feet. Ans. 17° 11′ 19′′. 39. Find the length of the arc of 75° in the circle whose radius is 5 feet. Ans. 6.545 feet. 40. Find a side of the circumscribed equilateral triangle, the radius of the circle being R. 41. Ans. 2R 13. Find a side of the circumscribed regular hexagon, the radius of the circle being R. 2R Ans. 42. Find the length of the arc subtended by one side of a regular dodecagon in a circle whose radius is 12.5 feet. Ans. 6.545 feet. 43. Find the area of a sector of 60° in the circle whose radius is 10 inches. Ans. 52.3599 sq. ins. 44. The area of a given sector is equal to the square constructed on the radius: find the number of degrees in the arc of the sector. Ans. 114° 35′ 29′′.6. 45. Find the area of the segment of 60° in the circle whose radius is 2 feet. Ans. 0.362344 sq. ft. 46. Find the radius of the circle in which the sector of Ans. .564 inches. 45° is .125 square inches. 47. Two tangents make with each other an angle of 60° required the lengths of the arcs into which their points of contact divide the circle, the radius being 21 inches. Ans. 44 inches, 88 inches. 48. Venice is due south of Leipsic 5° 55': how many miles apart are they, the radius of the earth being 4000 miles? Ans. 413 miles. 49. The three sides of a triangle are 9, 10, 17 inches respectively: find (1) its area, and (2) the area of the inscribed circle. Ans. (1) 36 sq. ins.; (2) 47. See (398), Ex. 3. 50. Mount Washington is visible from a point at sea 87 miles off: required the height of the mountain. PROBLEMS. Ans. 6270 feet. 51. To circumscribe a square about a given circle. 52. To inscribe a regular octagon in a given square. 53. Given the base of a triangle, the difference of its sides, and the radius of the inscribed circle: construct the triangle. From the given base AB, cut off AD = given difference; bisect DB in E, draw EO to AB and the given rad., etc. 54. Describe a circle which shall touch a given circle and two given straight lines which themselves touch the given circle. 55. In a given circle inscribe a triangle whose angles are as the numbers 2, 5, 8. See (453). 56. To inscribe a regular hexagon in a given equilateral triangle. 57. To construct a circle equivalent to the sum of two given circles. 58. In a given equilateral triangle, construct three equal circles tangent to each other and to the sides of the triangle, and find the radius of these circles in terms of the side of the triangle. 59. Construct an isosceles triangle having each angle at the base double the third angle. 60. Given the vertical angle of a triangle and the radius of the circumscribed circle: find the locus of the centre of the inscribed circle. Describe a O ABC with the given radius, draw AB cutting off segment ACB containing the given ; bisect arc AB opp. to C in D, draw any chord DC; bisect CAB by AO meeting CD in O, O is the centre of the inscribed : <DOA = ZOCA +20AC=2BAD+20AB = ¿DAO; .'. DO = DA; .'. etc. 61. Given a vertex of a triangle, the circumscribed cir |