always lies between p and p' (391), its difference from either must be less than p-p'; hence 2o. p and p' have each for limit c. Q.E.D. (235) Now when R2-2 becomes indefinitely small, P — P' becomes indefinitely small; and since always P>S, but P' < S, s lies always between P and P', (Ax. 8) (its difference from either being less than P — P';) .. P and P' have each for limit s. Q.E.D. EXERCISE 566. The apothem of an inscribed equilateral triangle is equal to half the radius of the circle. 567. The apothem of an inscribed regular hexagon is equal to half the side of the inscribed equilateral triangle. 568. Any arc is greater than its chord, but less than the sum of the tangents drawn from a point to its extremities. 569. Every equilateral polygon inscribed in a circle is also equiangular. 570. Every equiangular polygon inscribed in a circle is also equilateral, if the number of sides is odd. 571. Every equilateral polygon circumscribed about a circle is also equiangular, if the number of sides is odd. 572. Every equiangular polygon circumscribed about a circle is also equilateral. 573. If the alternate vertices of a regular hexagon are joined by straight lines, show that the figure formed is a regular hexagon. 574. What ratio has the latter figure to the former ? PROPOSITION IX. THEOREM. 393. Circumferences are to each other as their radii. Given: Two circumferences, C c', with radii R, R', C: C = R: R'. Conceive regular polygons of n sides to be inscribed in each of the circumferences (381), and by continual doubling (382), the number of sides to become indefinitely great. Let p, p', denote the variable perimeters. Now the limits of these equal variables are equal, c being the limit of p and c' of p' (392). Hence 394. COR. 1. Circumferences are to each other as their diameters. For C: C'R : R' ; .. C: C' 2R: 2 R' = D : D'. = (393) (253) 395. COR. 2. The ratio of the circumference to the diame ter is constant. For C: CD: D'; .. C: DC': D'. (394) 396. SCHOLIUM. The numerical value of this constant ratio, that is, the number showing how many times a circumference contains its diameter, is denoted by the Greek letter π. It is an incommensurable number, but its value, as will presently be shown, can be obtained to any required C C degree of precision. Since = =, we have the imporD 2 R tant relations, 397. A circle is equivalent to one half the rectangle contained by its radius and circumference. E B Given: C, the circumference, and R, the radius, of a circle ABE; To Prove Circle ABE is equivalent to rectangle R C. Let r denote the apothem, and p the perimeter, of a regular polygon P inscribed in ABE. Then P 1⁄2rect. r · p. (387) Conceive the number of sides by continual duplication to become indefinitely great. Then P has for limit for limit rect. RC, C (392) ...O ABErect. R. C, Q.E.D. (236) (they being the limits of variables always equal.) Geom. - 14 SCHOLIUM. This theorem may be stated under the form: The area of a circle is measured by one half the product of its radius and perimeter. 398. COR. 1. The area of a circle is equal to π times the square of its radius. For denoting the area of the circle by s, since S = R. C, (397), and C = 2π · R (396), S= RX 2π• R = π R2. 399. COR. 2. The areas of circles are to each other as the squares of their radii. 400. DEFINITION. A segment of a circle is the figure bounded by an arc and its chord. 401. DEFINITION. A sector of a circle is the figure bounded by two radii and their intercepted arc. 402. DEFINITION. Similar segments and sectors in different circles are such as have arcs measuring equal angles at the center. 403. COR. 3. The area of a sector is measured by one half the product of its radius and arc. For the sector is to the circle as the arc of the sector is to the circumference. 404. COR. 4. Similar sectors are as the squares of their radii. For they are like parts of their respective circles. EXERCISE 575. What figure is both a segment and a sector? 576. The area of one circle is twice that of another. ratio of their radii ? What is the 577. The radii of two similar segments is as 3 to 5. What is the ratio of their areas ? PROPOSITION XI. THEOREM. 405. Similar segments are to each other as the squares of their radii. B B' Given: OA = R, O'A' = R', radii of similar segments B, B'; To Prove: B: B' R': R'2. = Since 200', (Hyp.) sector OABC is similar to sector O'A'B'C'. (402) Since OA = OC, o'A' = o'c', and ≤ 0 = ≤ 0', AOAC is similar to ▲ O'A'C'. (290) Since sect. OABC: sect. O'A'B'C' = R2: R12, (404) and ▲ 04C': ▲ O'A'C' = 0.4′ : 0'A'2 = R2: R'2, (342) sect. OABC: ▲ OAC = sect. O'A'B'C': ▲ o'A'c'; (232"'') .. OABC- OAC: OAC = O'A'B'C' :. seg. B:seg. B' = ▲ OAC : ▲ O'A'C': o'A'c': o'A'c'; (247) R2: R2. Q.E.D. = (244) EXERCISE 578. In the diagram for Prop. XI., what must be the ratio of OA to O'A' if segment B is of segment B'? 579. In the same diagram, if the segment is of its circle, how many degrees are there in arc AC? 580. In the same diagram, if A were joined with the mid point D of arc AC, and Z CAD were found to be 38°, how many degrees in LAOC? 581. In the same diagram, if a line AF were drawn perpendicular to OA, and CAF were found to be 41°, how many degrees in arc AC? How many, in the angle formed by lines drawn from A and C to any point in arc AC? |