Convex pentagon ABCDE is inscribed in circle W such that AC/DE=2/3, AE=CD, and AB=BC. Suppose the distance from B to line AC is 144 and the distance from B to
line DE is 864. Find the radius of W.
Lines BW and AC intersect at F. FC=2x.
Lines BW and DE intersect at G. GD=3x.
Note ACDE is an isosceles trapezoid. There are two cases to consider: This trapezoid contains W or it doesn't.
Case 1: W is not inside ACDE.
On right triangles DGW and CFW
GW=r-864, DG=3x, DW=r
FW=r-144, CF=2x, CW=r
It's a simple task to solve the system of two pythagorean theorems to find r=648.
Case 2 W is inside ACDE
The only difference is GW=864-r
but GW^2 will be the same as Case 1 so the solution is the same r.
incidentally x=144sqrt2
Also, the numbers given in this problem are surprisingly big given they all have a GCD of 144.
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Posted by Jer
on 2024-11-03 12:54:04 |
Solution
