346. THEOREM. Two regular polygons of the same number of sides are similar. Outline of Proof: (1) Show that all pairs of corresponding angles are equal. (2) Show that the ratios of pairs of corresponding sides are equal. Hence the polygons are similar (Why?). 347. THEOREM. The perimeters of two regular polygons having the same number of sides are in the same ratio as their radii or their apothems. Outline of Proof: Show (1) that each triangle formed by a side and two radii in one polygon is similar to the corresponding triangle in the other polygon. (2) That AB r = α = where AB and A'B' are the two a' sides, r and r' the corresponding radii and a and a' the corresponding apothems. And so for the remaining pairs Draw the figure and give the proof in full. = 348. THEOREM. The areas of two regular polygons of the same number of sides are in the same ratio as the squares of the corresponding radii or apothems. Outline of Proof: Divide the polygons into pairs of corresponding triangles as in the preceding proof. (1) Show that triangles. (2) Hence p2 ΔΙ a2 a'2' and so for each pair of ΔΙ+ΔΙΙ+ΔΙΙ+ ▲ I' + ^ II' + ▲ III' + Draw figure and give the proof in full. /2 PROBLEMS AND APPLICATIONS. 1. Find the ratio of the perimeters of squares inscribed in and circumscribed about the same circle. 2. Find the ratio of the perimeters of regular hexagons inscribed in and circumscribed about the same circle. 3. Find the ratio between the perimeters of regular triangles inscribed in and circumscribed about the same circle. 4. Find the ratio of the areas of regular triangles inscribed in and circumscribed about the same circle. Also find the ratio of the areas of such squares and of such hexagons. 5. The perimeter of a regular hexagon inscribed in a circle is 24 inches. Find the perimeter of a regular hexagon circumscribed about a circle of twice the diameter. 6. The area of a regular triangle circumscribed about a circle is 64 square inches. What is the area of a regular triangle inscribed in a circle of one third the radius? 7. The area of a regular hexagon inscribed in a circle is 48 square inches. What is the area of a regular hexagon circumscribed about a circle whose diameter is 13 times that of the first? 8. A chord AB bisects the radius perpendicular to it. Find the central angle subtended by the chord. State the result as a theorem. 9. State and prove the converse of the theorem in Ex. 8. 10. Find the area of a regular triangle inscribed in a circle whose radius is 6 inches. 11. Find the area of a regular triangle inscribed in a circle whose radius is r inches. 12. One of the acute angles of a right triangle is 60° and the side adjacent to this angle is r inches long. Find the remaining sides of B the triangle. 13. A regular triangle is circumscribed about a circle of radius r. Find its area. SUGGESTION. First find DC in the figure. 14. A regular triangle of area 36 square inches is inscribed in a circle. Find the radius of the circle. 15. Find the radius of a circle if the area of its regular inscribed triangle is a. 16. Find the radius of the circle if the area of the regular circumscribed triangle is a. 17. Find the radius of a circle if the difference between the perimeters of the regular inscribed and circumscribed triangles is 12 inches. 18. Find the radius of a circle if the difference of the perimeters of the regular inscribed and circumscribed hexagons is 10 inches. 19. If the area of a circumscribed square is 25 square inches greater than that of an inscribed square, what is the diameter of the circle? 20. Find the radius of a circle if the difference between the areas of the inscribed and circumscribed regular triangles is 25 square inches. 21. Find the radius of a circle if the difference between the areas of the regular inscribed and circumscribed hexagons is 25 square inches. 22. The difference between the areas of the squares circumscribed about two circles is 50 square inches and the difference of their diameters is 4 inches. Find each diameter. 23. If the inscribed and escribed circles and O' of an equilateral triangle are constructed as shown in the figure, find the ratio of their radii. Does this ratio depend upon the size of the triangle? 24. Given a triangle ABC and a segment a, show how to construct a segment DE || AB and equal to the segment a, such that the points D and E shall lie on the sides CA and CB respectively or on these sides extended. 25. A triangular plot of ground ABC is to be laid out as a triangular flower bed with a walk of uniform width extending around it. (a) Prove that the flower bed is similar to the original triangle. (b) Show that the corners of the flower bed lie on the bisectors of the angles of the original triangle. D Ꭺ B C S B to S and parallel to AB, the points A' and B' lying on the bisectors of the angles A and B respectively. See Exs. 16–19, page 172. (d) Draw A'C' || AC and B'C' || BC. Prove that the area of the flower bed A'B'C', as thus constructed, is equal to the area of the walk. (e) Construct the figure for the flower bed so that its area is five times that of the walk. 26. Given a rectangular plot of ground. Is it always possible to lay off on it a walk of uniform width running around it so that the plot inside the walk shall be similar to the original figure? Prove. 27. Is the construction proposed in Ex. 26 always possible in the case of a square? of a rhombus ? Prove. 28. If a side of a regular hexagon circumscribed about a circle is a, find the radius of the circle. 29. On a regular hexagonal plot of ground whose side is 12 feet a walk of uniform width is to be laid off around it. Find by algebraic computation the width of the walk if it is to occupy one half the whole plot. 30. Find by geometric construction the width of the walk in Ex. 29. 31. Show that the figure inside the walk in Ex. 29 is a regular hexagon. AE 3 32. Given a segment AB, find three points C, D, E on it so that, AC2 1 AD2 1 AB2 4 AB2 2' AB2 4 and = = E D 33. A regular hexagon ABCDEF is to be divided into four pieces of equal area by segments drawn parallel to its sides forming hexagons as shown in the figure. If AB=24 feet, find a side of each of the other hexagons and also the apothem of each. IV F 34. Without computing algebraically the apothem or sides of the inner hexagons in Ex. 33, show how to construct the figure geometrically. See Ex. 6, § 331. Also Ex. 18, page 172. Use § 348. 35. In a given hexagonal polygon whose side is a find in terms of a and n the width of a walk around it which will occupy one nth of the area of the whole polygon. 36. By means of hexagons similar to those in Ex. 33, divide a given regular hexagon into three parts such that the outside part is the whole area and the next of the whole area. 37. Compute the sides and the apothem of the two hexagons constructed in Ex. 36 if the side of the given hexagon is 12 inches. 38. In the adjoining pattern find two regular hexagons whose areas are in the ratio 1:4 and show that this agrees with theorem, § 348. 39. If a polygon is circumscribed about a circle, show that the bisectors of all its angles meet in a point. 40. Given any polygon circumscribed about a circle. Within it draw segments parallel to each of its sides and at the same distance from each side. Show that these segments form a polygon similar to the first. 41. If the bisectors of all the angles of a polygon meet in a point prove that a circle may be inscribed in it (tangent to all its sides). What is the relation of the theorems in Exs. 39 and 41? L E B H B 42. Given a polygonal plot of ground (boundary is a polygon) such that a path of equal width on it around the border leaves a polygonal plot similar to the first. Prove that a circle may be inscribed in the polygon which forms the boundary of either plot. 43. In the figure, ABCD is a square and EFGHKLMN is a regular octagon. |