MathDB
Regional Area

Source: AIME 2010I Problem 13

March 17, 2010
geometryrectangleratiotrapezoidtrigonometryanalytic geometrygraphing lines

Problem Statement

Rectangle ABCD ABCD and a semicircle with diameter AB AB are coplanar and have nonoverlapping interiors. Let R \mathcal{R} denote the region enclosed by the semicircle and the rectangle. Line ℓ \ell meets the semicircle, segment AB AB, and segment CD CD at distinct points N N, U U, and T T, respectively. Line ℓ \ell divides region R \mathcal{R} into two regions with areas in the ratio 1:2 1: 2. Suppose that AU \equal{} 84, AN \equal{} 126, and UB \equal{} 168. Then DA DA can be represented as mn m\sqrt {n}, where m m and n n are positive integers and n n is not divisible by the square of any prime. Find m \plus{} n.
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