Plane and solid geometry . Given AB, an arc of any bisect AB, The construction, proof, and discussion are left as anexercise for the student. Ex. 429. Construct an arc of 45°; of 30°. Construct an arc of 30°,using a radius twice as long as the one previously used. Are these two30° arcs equal ? Ex. 430. Distinguish between finding the mid-point of an arc andthe * center of an arc. BOOK II 123 Proposition VII. Theorem 307. In equal circles, or in the same circle, if two chordsare equal, they are equally distant froin the center; con-versely, if two chords are equally distant from the c
Plane and solid geometry . Given AB, an arc of any bisect AB, The construction, proof, and discussion are left as anexercise for the student. Ex. 429. Construct an arc of 45°; of 30°. Construct an arc of 30°,using a radius twice as long as the one previously used. Are these two30° arcs equal ? Ex. 430. Distinguish between finding the mid-point of an arc andthe * center of an arc. BOOK II 123 Proposition VII. Theorem 307. In equal circles, or in the same circle, if two chordsare equal, they are equally distant froin the center; con-versely, if two chords are equally distant from the center,they are I. Given circle 0 with chord AB = chord CD, and let OE andOF be the distances of AB and CD from center 0, respectively. To prove OE = OF. Argument 1. Draw radii OB and OC, 2. E and F are the mid-points of AB and CDy respectively. 3. . rt. A OEB and OCF, EB = CF. 4. OB = OC, 5. .-. A OEB = A OCF. 6. .-. 0E= OF, II. Conversely: Given circle 0 with OE, the distance of chord AB from center0, equal to OF, the distance of chord CD from center 0. To prove chord AB = chord CD,Hint. Prove A OEB = A OCF. Reasons 1. § 54, 15. 2. § 302. 3. § 54, 8 a. 4. § 279, a, 5. § 211. 6. § 110. Ex. 431. If perpendiculars from the center of a circle to the sides ofan inscribed polygon are equal, the polygon is equilateral. Ex. 432. If through any point in a diameter two chords are diawnmaking equal angles with the diameter, the two chords are equal. 124 PLANE GEOMETRY Proposition VIII. Theorem 308. Lv equal circles, or in tlve same circle, if twochords are unequal, the greater chord is at the
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