Name: james
Who is asking: Student
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 
or m/n = C = 360/49 and hence m+n = 409.
Cheers,
Haragauri
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