Question 1175957
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Let the vertices of the triangle be A(0,0), B(12,0), and C(0,5).<br>
Let a, b, and c be the radii of the circles centered at A, B, and C respectively.<br>
Side AB has length 12 and side AC has length 5; the hypotenuse BC has length 13.<br>
Then, since the circles are externally tangent to each other,
a+b=12
a+c=5
b+c=13<br>
This system of equations is easy to solve by first adding all three equations together to get<br>
2a+2b+2x=30
a+b+c=15<br>
Then comparing that equation to the earlier three equations gives us a=2, b=10, and c=3.<br>
Use the formula for the area of a circle to complete the problem.<br>