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 $a - b.$
Right scalene triangle.
Sides: a = 10 b = 24 c = 26
Area: T = 120
Perimeter: p = 60
Semiperimeter: s = 30
Angle ∠ A = α = 22.62° = 22°37'11″ = 0.395 rad
Angle ∠ B = β = 67.38° = 67°22'49″ = 1.176 rad
Angle ∠ C = γ = 90° = 1.571 rad
Height: ha = 24
Height: hb = 10
Height: hc = 9.231
Median: ma = 24.515
Median: mb = 15.62
Median: mc = 13
Inradius: r = 4
Circumradius: R = 13
Area of circumcircle =13^2 x pi
Area of incircle =4^2 x pi
[13^2 x pi] - [4^2 x pi] =153pi
Yes, that is the correct answer.
My explaination is
we know that Triangle ABC is a right triangle because we know the Pythagorean Theorem's converse is true.
The hypotenuse of a right triangle is a diameter of the triangle's circumcircle.
Therefore, the area of the circumcircle shoud be 13^2*pi = 169pi
To find the incircle's area, we see $[ABC] = (10)(24)/2 = 120$
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 $169\pi - 16\pi = \boxed{153\pi}$
So whoever discredited the guest who got the correct answer with a great explanation, is wrong, and
Melody is absolutly correct although that guest may not be the same guest who asked the problem. :p
So yes, the answer should be 153pi.
