Date: Sun, 11 Jan 1998 11:25:32 -0600 (CST)

Name: james

Level: Secondary

Question:

A hexagon inscribed in a circle has three consecutive sides each of length 3 and three consecutive sides each of length 5. The chord of the circle that divides the hexagon into two trapezoids, one with three sides each of length 3 and the other with three sides each of length 5, has length equal to m/n, where m and n are relatively prime positive integers. Find m+n.
(a) 309 (b) 349 (c) 369 (d) 389 (e) 409

Hi James,

 In Figure 1 X is the center of the circle. Let C = m/n be the length of PS. Since the chord TU subtends the angle theta at the center and the angle UPT at the circumference, angle UPT = theta/2. Since |TS|=|UT|=3, angle TPS is also theta/2. Hence angle UPS = theta. By a simular argument angle SPQ = phi. Figure 1 In Figure 2 R is the radius of the circle. Thus, since angle YXU = theta/2, |YU|/|UX|=sin(theta/2) and thus 3 = 2 R sin(theta/2). Simularly using triangle XQR 5 = 2 R sin(phi/2). Thus  - - - Equation 1. Figure 2 In Figure 3 |PA|/|UP|=cos(theta) and thus |PA|=3 cos(theta). Simularly |BS|=3 cos(theta) and hence C=|PS|=3+6cos(theta). Using the fact that we get  - - - Equation 2. Simularly  - - - Equation 3. Figure 3
Thus, using equations 1, 2 and 3 we get

or m/n = C = 360/49 and hence m+n = 409.

Cheers,
Haragauri

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