Problem: Sides AB,BC and CD of (simple) quadrilateral ABCD have lengths 4,5 and 20, respectively. If vertex angles B and C are obtuse and sinC=−cosB=53, then side AD has length
Answer Choices:
A. 24
B. 24.5
C. 24.6
D. 24.8
E. 25
Solution:
Let E be the intersection of lines AB and CD, and let β and θ be the measures of ∠EBC and ∠ECB, respectively. Since
cosβ=−cosB=sinC=sinθβ+θ=90∘.∠BEC is a right angle, and BE=BCsinθ=3;CE=BCsinβ=4
Therefore, AE=7.DE=24 and AD, which is the hypotenuse of right triangle ADE, is 25.