339. Cor. II. If the arcs subtended by a regular polygon of n sides be bisected, what will the chords. of these arcs form? Suppose A B C D E and A' B'C' D' E' are regular polygons of the same number of sides. (1) Compare the sum of all thes in P and P. (2) Compare ▲ A and ▲ A', ▲ B and ▲ B'. Why does A = [ A? / B = ≤ B'? ̧ Compare A B, B C, CD, etc. Compare A'B', B'C', C'D', etc. Can you show that P and P' are similar? Call this Prop. III. 341. PROPOSITION IV. Prove that regular polygons of the same number of sides may be divided into the same number of similar As, similarly placed. 342. PROPOSITION V. C M F In A-D and A'-D' we have two regular polygons of the same number of sides. Let P and P' be the perimeters and S and S' be the areas of the figures. Let O and O' be the centers. (How find them?) Join O A, O B, O' A', O' B', and draw the Ls O F and O' F' to A B and A'B' respectively. Call O A, R; O' A', R'; O F, r; O' F', '. Compare Zs A OB and A' O'B'. Can you compare the ratio of O A and O' A' with ratio of O B and O' B'? What conclusion concerning the As? Compare the ratio of A B and A' B' with the ratio of O F and O' F? Substitute for O B, O' B', O F, O'F their values in the last comparison. 343. PROPOSITION VI. Show how to find the area of any regular polygon. 344. PROPOSITION VII. Show how to inscribe a square in a given O. 345. PROPOSITION VIII. Can you prove how to inscribe a regular hexagon in a circle? EXERCISES. 328. In how many ways can you construct an equilat eral triangle? 329. If v = radius of the O, and s the side of the inscribed equilateral ▲, show that s=r v3. 330. The distance from the center of an inscribed equilateral to a side is γ 331. Inscribe an equilateral A and a regular hexagon in the same. Compare their areas. in Ex 332. Circumscribe an equilateral about the ercise 331. Compare the hexagon with the 2 As. State your conclusion in neat form. In this figure let A' B' be one side of a regular circumscribed polygon of n sides. Call its perimeter P, and let O be the center of the polygon and circle. Draw O A', O B' cutting the arc at A and B. Join A and B. Prove A B || to A' B'. Can you show that A B is the side of a regular inscribed polygon of n sides? Call its perimeter p. How often Show that How many Join A F, B F and draw tangents at A and B. is A B used in the regular inscribed polygon? A F is the side of a regular inscribed polygon. sides? Can you show that M N is the side of a regular cir. cumscribed polygon? How many sides to the polygon of which M N is a side? Call the perimeter whose P', and the regular polygon whose side is A F, p'. side is M N, Show that P A' B' Write the values of A B, MN, A F. Join O M, n A' M O F. Prove that M O bisects / A' O F. Show that M F O A' O F What are O A' and O F of the polygons? How are the radii of regular polygons of the same number of sides related? (§342.) What axiom shows this, P A'M, P M F Rewrite this equation by composition. What is the sum of A'M and M F? How is M F related to M N? P+P What is P'? Express this equation in words. Can you prove As A B F and A M F similar? Show A BM N. Substitute the values of A F, p A B, M N, and show that (2) ' = vp · P'. By means of (1) and (2) the perimeter of regular inscribed and circumscribed polygons of double the number of sides may be found, for any values of P and p. Call the above Prop. IX. Write a statement of it. EXERCISES. 333. The side of an inscribed sq. = r √ē = apothem, and S area, and the radius of the = 1, prove that: (a) In a regular inscribed octagon s= V2 — √2, r = 2 √2 + √ 2, S = 2 VT. (6) In a regular circumscribed octagon s = 2 √2 — 2, R = V4 — 2 √2, S = 8 √2 — 8. r = (c) In a regular inscribed dodecagon s= (d) In a regular circumscribed dodecagon s=4-2√3, RV8 — 4 √3, S — 24 — 12 √3. 335. In a given sector whose at the center = 90° inscribe a square. Prove the area = R2 |