BOOK V Ex. 1. Prove that the diagonals drawn from one vertex of a regular polygon having n vertices to each of the other vertices divides the angle at that vertex into (n − 2) equal parts. Ex. 2. Prove that the central angle of any regular polygon is the supplement of the vertex angle of the polygon. Ex. 3. Prove that the sum of the perpendiculars drawn from any point within a regular polygon to the sides of the polygon is equal to the apothem multiplied by the number of sides of the polygon. Suggestions.-Connect the point with each vertex. Notice that the sum of the triangles so formed equals the polygon. Express the area of each triangle and form an equation. Ex. 4. In the figure for § 365 prove that: (a) 84 > 88 > 816, etc. (See § 362.) (b) asas < a16, etc. (c) k4 <ks <k16, etc. Ex. 5. Prove that an equiangular polygon inscribed in a circle is regular if the number of sides is odd. Ex. 6. Prove that an equiangular polygon circumscribed about a circle is regular. Suggestions.-1. Draw the chords joining the points of tangency. 2. Prove the resulting As: (a) are isosceles; (b) are mutually equiangular; (c) that XY YZ, etc. See diagram in § 367. Complete the proof. Ex. 7. Repeat Ex. 14, p. 227, for a regular octagon circumscribed about a circle of radius 10. Ex. 8. Prove that diagonal AE of regular octagon ABCDEFGH is the perpendicular-bisector of diagonal BH. Ex. 9. In the adjoining figure, ABCD and EFGH are squares inscribed in the circle, such that AF = FB BG, etc. Is RSTUVWXY a regular octagon ? Ex. 10. Construct a Maltese cross having the dimensions indicated. Ex. 11. Prove that the construction indicated in the adjoining figure serves to inscribe a regular octagon in the square. Ex. 12. A regular octagon is inscribed in a circle of radius 10. Compute s8, ps, as, and kg. Ex. 13. Prove that for a regular octagon inscribed in a circle of radius R: Ex. 14. Construct a regular octagon having its sides 1 inch long. Ex. 15. What is the relation between the area of the inscribed and of the circumscribed equilateral triangles of a given circle ? Ex. 16. What is the relation between the perimeter of the inscribed and of the circumscribed equilateral triangles of a given circle? Ex. 17. A regular hexagon is inscribed in a circle of radius r. Prove: r√3 (a) s6 = r; (b) a6 = ; (c) P6 = 6 r ; (d) ke: = 3r2√3 Ex. 18. A regular triangle is inscribed in a circle of radius r. Prove: 1 32√3. 2 4 (a) 83 = r√3; (b) as = =¦r; (c) p3=3r√3; (d) k3 = Ex. 19. Prove that the apothem of an equilateral triangle is one third the altitude of the triangle. Ex. 20. (a) In a circle of radius 2.5 in., inscribe a regular hexagon. (b) Also inscribe in the same circle a regular triangle and a regular 12-gon. (c) Prove that 83 > 86 > 812, etc. Ex. 21. What is the perimeter and area of a regular circumscribed hexagon about a circle of radius 10? Ex. 22. Repeat the foregoing exercise for a circle of radius R. Ex. 23. What is the perimeter and the area of a regular triangle circumscribed about a circle of radius 10? Ex. 24. Repeat the foregoing exercise for a circle of radius R. Ex. 25. Prove that the diagonals AC, BD, CE, etc., of regular hexagon ABCDEF form another regular hexagon. Suggestion.-Prove that a circle can be inscribed in the inner hexagon. Ex. 26. Prove that the area of the inner hexagon of the foregoing exercise is one third the area of ABCDEF. Suggestion.-Express the area of each polygon in terms of the radius OB of ABCDEF. B. C H G K A D N M E Ex. 27. Prove that the area of a regular inscribed hexagon is a mean proportional between the áreas of an inscribed and of a circumscribed equilateral triangle. Suggestion.-Express the areas of each in terms of the radius. Ex. 28. In a given equilateral triangle, inscribe a regular hexagon having two of its vertices lying on each side of the triangle. Ex. 29. Construct a regular hexagon having given one of the diagonals joining two alternate vertices. Ex. 30. A square is inscribed in an equilateral triangle whose side is a, having two vertices in one side of the triangle, and one in each of the other sides. Compute the area of the square. Ex. 31. A regular 12-gon is inscribed in a circle of radius R. Prove: (a) 812 = RV2 - √3; R√2+√3; (c) P12 12 RV 2 −√3; Ex. 32. If the diagonals AC and BE of regular pentagon ABCDE intersect at F, prove that BE = AE + EF. Ex. 33. Prove that the figure FGHKL formed by parts of the diagonals of regular inscribed petagon ABCDE is also a regular pentagon. (See figure on page 302.) B Suggestions. Prove that a circle can be inscribed in FGHKL. Ex. 34. In the figure of Prop. VIII, Book V, prove that OM is the side of a regular pentagon inscribed in the circle which can be circumscribed about Δ ΟΒΜ. Suggestion. - How large is / OBM? A H K E Ex. 35. Construct a regular pentagon having given one of its sides. Ex. 36. Construct a regular pentagon having given one of its diagonals. Ex. 37. If R represents the radius of the circle circumscribed about a regular decagon, prove: Ex. 38. Find the area of the circle inscribed in a square whose area is 25. Ex. 39. If the radius of a circle is 3√3, what is the area of the sector whose central angle is 150° ? Ex. 40. Find the radius of the circle equal to a square whose side is 10. Ex. 41. Find the radius of the circle whose area is one half the area of the circle whose radius is 15. Ex. 42. Find the area of the square inscribed in the circle whose area is 196 π sq. in. Ex. 43. The area of one circle is 25 the area of another. Find the radius of the second if the area of the first is 15. Ex. 44. The side of a square is 8. Find the circumference of its inscribed and circumscribed circles. Ex. 45. The side of an equilateral triangle is 6. Find the area of its inscribed and circumscribed circles. Ex. 46. The area of a regular hexagon inscribed in a circle is 24 √3. What is the area of the circle? Ex. 47. If the apothem of a regular hexagon is 6, what is the area of its circumscribed circle ? Ex. 48. Two plots of ground, one a square and one a circle, each contain 70686 sq. ft. How much greater is the perimeter of the square than the length of the circle? Ex. 49. The perimeter of a regular hexagon circumscribed about circle is 12√3. What is the circumference of the circle? Ex. 50. The length of the arc subtended by the side of a regular inπ in. What is the area of the circle? scribed 12-gon is 1 3 Ex. 51. If the length of a quadrant is 1, what is the diameter of the circle ? Ex. 52. Prove that the area of the square inscribed in a sector whose central angle is a right angle, is equal to one half the square on the radius. Ex. 53. If a circle is circumscribed about a right G triangle, and on each of the legs of the triangle as diameters semicircles are drawn, exterior to the triangle, the sum of the areas of the crescents thus A formed equals the area of the triangle. Prove area AECG + area BFCH area ▲ ABC. = Suggestion. From the sum of ▲ ABC and the semicircles on AC and BC, subtract the semicircle on AB. Express each area in terms of sides a, b, and c of the triangle. Ex. 54. Construct three equal circles having the vertices of an equilateral triangle as their centers and for their radii one half the side of the triangle. Compute the area of that part of the interior of the triangle which is exterior to each of the circles, if the length of the side of the triangle is s. Ex. 55. Upon a segment AC draw a semicircle. Upon AC locate a point B, not the center of AC. Upon AB and BC as diameters draw semicircles within the one drawn upon AC as diameter. Prove that the area of the surface lying within the largest semicircle and exterior to the smaller ones equals the area of the circle drawn upon BD as diameter, where BD is the perpendicular to AC at B meeting the largest semicircle at D. (Due to Archimedes.) Ex. 56. With the vertices of an equilateral triangle as centers and the side of the triangle as radius, three equal circles are drawn. Determine the area of that figure which is common to the three circles. a |