In the figure, triangle ABC is inscribed in a semicircle with diameter AC of length 20 inches, and AB = 12 inches. When the area of the shaded region, in square inches, is expressed in the form \(a\pi - b\) what is the value of a+b.

https://latex.artofproblemsolving.com/f/3/e/f3ed2bd688e09c00947e4d6cdb13abeb62fed45d.png

supremecheetah May 23, 2023

#1**0 **

Since AC is the diameter of the semicircle, O is the center of the semicircle. Since AB is a diameter of the semicircle, ∠AOB=90∘. Therefore, triangle AOB is a right triangle with legs of length 10 and 12, so its area is (1/2)(10)(12)=60. The area of the semicircle is (1/2)(202)(pi)=200pi. The area of the shaded region is the difference between the area of the semicircle and the area of triangle AOB, or 200pi−60=140pi−60.

Therefore, a + b = 140 + 60 = 200.

Guest May 23, 2023