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A triangle has side lengths of 10, 24, and 26.  Let a be the area of the circumcircle, and let b be the area of the triangle.  Compute a - b in terms of pi.

 Jan 7, 2022
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My best shot:

Okay so i don't know how to explain this, but the hypotenuse of this triangle is the diameter of the circle, so the circle area is 

\(\pi\times r^2\)

\(169\pi\)

the area of the triangle is 

\(\frac{10*24}{2}\)

\(120\)

and $s = (10+24+26)/2 = 30$, so $r = [ABC]/s = 120/30 = 4$.

When $[ABC] = rs$, where $r$ is the inradius and $s$ is the semiperimeter of the triangle.

Therefore, the area of the incircle is $4^2\pi = 16\pi$.Finally, the area of the circumcircle is $169\pi - 16\pi = \(\boxed{153\pi}$\)

 Jan 7, 2022

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