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 $a - b.$

Guest May 9, 2020

#1**+1 **

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**

Guest May 9, 2020

#4**+1 **

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.

Guest May 10, 2020

edited by
Guest
May 10, 2020