250. Corollary 2. If tangents are drawn at the vertices of a regular inscribed polygon, they form a regular circumscribed polygon of the same number of sides. The proof is left to the student. EXERCISES 1. The perimeter of a regular inscribed polygon is less than that of a regular inscribed polygon of twice as many sides. 2. The perimeter of a regular circumscribed polygon is greater than that of a regular circumscribed polygon of twice as many sides. 3. Tangents drawn at the middle points of the arcs subtended by the sides of a regular inscribed polygon form a regular circumscribed polygon of the same number of sides. 4. The diagonals AC, BD, CE, etc., of a regular hexagon ABCDEF form another regular hexagon MNOPQR. SUGGESTION. - Prove diagonals AC, BD, etc., equal. Then prove that a circle can be inscribed in hexagon MNOPQR. 5. The diagonals of a regular pentagon ABCDE form another regular pentagon MNOPQ. 6. An equiangular polygon inscribed in a circle is regular if the number of sides is odd. 7. An equiangular polygon circumscribed about a circle is regular. J P 0 D K H E C Q N A B F M G E D P 251. Theorem. - Two regular polygons of the same number of sides are similar. Hypothesis. ABCD... and MNOP ... are regular polygons Proof. 1. ABCD... and MNOP ... are regular polygons of the same number of sides. Нур. 2. ... ∠A = ∠B = etc., and ∠ M = ∠ N= etc. Def. reg. poly. 3. ∠ A + ∠ B + etc. = ∠M+N+etc. §101 and Ax. I 4. ... if n is the number of sides of each polygon, 7. AB=BC= etc., and MN=NO = etc. Def. reg. poly. 252. Corollary. - The areas of two regular polygons of the same number of sides are to each other as the squares of their sides. The proof is left to the student. 253. Theorem. The perimeters of two regular polygons of the same number of sides are to each other as their radii, or as their apothems. Hypothesis. AB and CD are sides, and Mand N the centers, respectively, of two regular polygons of the same number of sides. Pand pare the perimeters, Randr the radii, and H and h the apothems, respectively, of the poly By aid of § 244, prove △ ABM~ △ CDN. 254. Corollary. - The areas of two regular polygons of the same number of sides are to each other as the squares of the radii, or of the apothems. The proof is left to the student. EXERCISES 1. Find the ratio of the perimeters and the ratio of the areas of two regular hexagons if their sides are 2 in. and 6 in., respectively. 2. Squares are inscribed in two circles of radii 2 in. and 8 in., respectively. Find the ratio of the perimeters of the squares and also of their areas. 3. What is the relation between the perimeters of the inscribed and circumscribed equilateral triangles of a circle? 4. What is the relation between the areas of the inscribed and circumscribed equilateral triangles of a circle? 5. Prove that the area of an inscribed square equals one half of the area of a circumscribed square of a circle. SUGGESTION. - Take the circumscribed square with its vertices on the radii produced of the inscribed square. Then OC = CB. Hence OB2 = 2 OC2 = 2 OA2. N A C B 6. Prove that the area of a regular inscribed hexagon equals three fourths of the area of a regular circumscribed hexagon of a circle. 255. Theorem. The area of any regular polygon equals one half of the product of its apothem and perimeter. Hypothesis. ABCD ... is a regular polygon with apothem h and perimeter p. Conclusion. Area of ABCD = hp. Proof. 1. h = apothem and p = perimeter of regular polygon ABCD .... Нур. 2. Draw radii OA, OB, OC, etc. 3. △ OAB = h × AB, △ OBC=1h × BC, etc. § 219 4. ... Δ ΟΑΒ +△ OBC+etc. = 1 h × AB+hxBC+etc. Ax. II 5. .·. Δ ΟΑΒ + △ OBC + etc. = 1 h (AB + BC + etc.). 6. ... Area of ABCD ... = hp. EXERCISES Factoring If R is the radius of a circle : 1. The side of an inscribed square equals R√2. 2. The area of an inscribed square equals 2 R2. 3. The side of an inscribed equilateral triangle equals R√3. 4. The area of an inscribed equilateral triangle equals & R2√3. 8. The area of a regular inscribed hexagon equals R2√3. 9. The side of a regular circumscribed hexagon equals R√3. 10. The area of a regular circumscribed hexagon equals 2 R2√3. 11. If the area of a square is 400 sq. in., find the apothem and radius, 12. If the area of an equilateral triangle is 12√3, find its radius. SUGGESTION. - R2√3 = 12√3. 13. If the area of an equilateral triangle is 108√3, find its apothem. If the radius of a circle is 24 in., find: 14. The area of an inscribed equilateral triangle. 15. The area of a circumscribed equilateral triangle. 16. The area of an inscribed square. 17. The area of a circumscribed square. 18. The area of an inscribed regular hexagon. 19. The area of a circumscribed regular hexagon. 20. The area of an inscribed regular hexagon is a mean proportional between the areas of the inscribed and circumscribed equilateral triangles. 21. If the diagonals joining the alternate vertices of a regular hexagon are drawn, the area of the second regular hexagon which they form is one third that of the original hexagon. : |