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

 Jun 11, 2020
 #1
avatar+8342 
+1

The triangle is a right-angled triangle.

 

Radius of circumcircle = 26/2 = 13

 

Area of circumcircle = \(\pi \cdot 13^2 = 169\pi\)

 

Radius of incircle = \(\dfrac{2\times\text{Area}}{\text{Perimeter}} = 4\)

 

Area of incircle = \(\pi \cdot 4^2 = 16\pi\)

 

Therefore \(a - b = 169\pi - 16\pi = 153\pi\)

 Jun 11, 2020
 #2
avatar+21955 
+1

Because this is a right triangle (102 + 242  =  262), the center of the circumcircle is the midpoint of the

hypotenuse. This means that the radius of the circumcircle = 26/2  =  13.

 

Because this is a right triangle, the radius of the incircle can be found using this formula:

   r  =  ( a + b - c ) / 2           where c is the hypotenuse and a and b are the other two sides.

 

From these radii, you can calculate the areas.

 Jun 11, 2020

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