posted by Jujube

1. A rhombus has a perimeter of 96, and the length of one of its diagonals is 32. The area of the circle inscribed in the rhombus can be expressed as k*pi/w where k and w are relatively prime positive integers. Find the value of k + w.

1. Reiny

the diagonals of a rhomus bisect each other at right angles, forming 4 equal right-angled triangles
the hypotenuse of each is 96/4 or 24 and one side is 32/2 or 16.
so by Pythagoras, the other side is sqrt(320)

The centre of the incsribed circle is at the intersection of the diagonals, so I drew a height inside one of the triangles.

by some basic geometry I found the r^2 = 1280/9

so the area of the circle is pi*1280/9 but that is given to be k*pi/w

so k*pi/w = pi*1280/9

since 1280/9 is relatively prime, k+w = 1280+9 = 1289

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