PROPOSITION IV. THEOREM 723. The perimeters of similar regular polygons are to each other as the radii of their inscribed or of their circumscribed circles; and the polygons are to each other as the squares of the radii. B R M S Let ABC F and MNR · ... ... s be two similar regular polygons. To Prove that their perimeters are proportional to the radii of the inscribed and of the circumscribed circles, and that their areas are proportional to the squares of these radii. Proof. Let x and y be the centers of the regular polygons. Draw xB and yN, and the apothems xE and yL. xB and yN are the radii of the circumscribed circles and XE and yL are the radii of the inscribed circles. 724. EXERCISE. S NR Ny Two squares are inscribed in circles, the diameters of which are 2 in. and 6 in. respectively. Compare their areas. 725. EXERCISE. A regular polygon, the side of which is 6 in., is circumscribed about a circle having a radius √3 in. Find the side of a similar polygon circumscribed about a circle the radius of which is 6 in. 726. EXERCISE. The perimeters of similar regular polygons are to each other as the diameters of their inscribed or of their circumscribed circles; and the polygons are to each other as the squares of the diameters. PROPOSITION V. PROBLEM 727. To inscribe a square in a given circle. B Let O be the center of the given circle. Required to inscribe a square in the circle. Prove ACBD an inscribed square. (§ 708.) Q.E.F. 728. COROLLARY I. Tangents to the circle at the extremities of the diameters AB and CD form a circumscribed square. 729. COROLLARY II. The side of the inscribed square is R√2. The side of the circumscribed square is 2R. The area of the inscribed square is 2R2. The area of the circumscribed square is 4R2. 730. COROLLARY III. By bisecting the arcs and drawing chords and tangents as described in § 710, regular polygons of 8, 16, 32, 64, etc., sides can be inscribed in and circumscribed about the circle. SANDERS' GEOM. - 15 731. EXERCISE. The radius of a circle is 5 ft. Find the side and the area of the inscribed square. 732. EXERCISE. Find the side and the area of a square circumscribed about a circle, having a diameter 6 in. long. 733. EXERCISE. The area of a square is 16 sq. in. Find the radius of the inscribed circle and also the radius of the circumscribed circle. PROPOSITION VI. PROBLEM 734. To inscribe a regular hexagon in a circle. D E B Let O be the center of the given circle. Required to inscribe a regular hexagon in the circle. Draw the radius OA. Lay off the chord AB=04. Draw OB. ▲ OAB is equilateral, and angle contains 60°. .. the arc AB is of the circumference, and the chord AB is one side of a regular hexagon. Complete the hexagon ABCDef. Q.E.F. 735. COROLLARY I. The chords joining the three alternate vertices form an inscribed equilateral triangle. 736. COROLLARY II. Tangents drawn at the vertices of the inscribed hexagon and of the triangle form a regular circumscribed hexagon and a regular circumscribed triangle. 737. COROLLARY III. If the arcs are bisected and chords and tangents are drawn according to § 710, regular polygons of 12, 24, 48, etc., sides will be inscribed in and circumscribed about the circle. 738. EXERCISE. The side of the inscribed equilateral triangle is R√3, and its area is R2√3. 739. EXERCISE. The side of the circumscribed equilateral triangle is 2 R√3, and its area is 3 R2√3. 740. EXERCISE. area is R2√3. 741. EXERCISE. The side of a regular inscribed hexagon is R, and its The side of a regular circumscribed hexagon is R√3, and its area is 2 R2√3. 742. EXERCISE. The area of a regular inscribed hexagon is double that of an equilateral triangle inscribed in the same circle. [Show this in two ways: 1st, by comparing the values of their areas as derived in §§ 738 and 740; 2d, by a geometrical demonstration using the figure of § 734.] 743. EXERCISE. What is the area of a regular hexagon inscribed in a circle, the radius of which is 4 in.? 744. EXERCISE. The area of a regular inscribed hexagon is 10 sq. in. What is the area of a regular hexagon circumscribed about the same circle ? 745. EXERCISE. The area of an equilateral triangle is 48 √3 sq. ft. Find the radii of the inscribed and of the circumscribed circles. 746. EXERCISE. The area of a regular hexagon is 54 a2 √3. Find the radii of the inscribed and of the circumscribed circles. 747. EXERCISE. Show that the circumscribed equilateral triangle is 4 times the inscribed equilateral triangle; that the circumscribed square is 2 times the inscribed square; and that the circumscribed regular hexagon is of the inscribed regular hexagon. 748. EXERCISE. compasses only. Divide a circumference into quadrants by the use of [SUGGESTION. The side of an inscribed square is the altitude of an isosceles triangle whose base is 2R and one of whose sides is R √3.] PROPOSITION VII. PROBLEM 749. To inscribe a regular decagon in a circle. B A Let O be the center of the given circle. Required to inscribe a regular decagon in the circle. Draw the radius 04. Divide it into extreme and mean ratio, OB being the greater segment. AOAC and BAC are similar. (§ 495.) 21=23+20 (?) or 1=2o. (?) 220+220 +20=180°. (?) .. ≤ 0 = 36°. .. the arc AC, the measure of ≤ 0, contains 36° of arc, and is of the circumference. The circumference can therefore be divided into ten parts, each equal to the arc 4C, and the chords joining the points of division form a regular inscribed decagon. Q.E.F. |