Question

In ABC, P is a point on AB such that ∠PCB = 45°, ∠B = 90°, and ∠ACB = 60°. Find the length of AP if BC = 10 cm.

Answer 1

Given

∠B = 90°, ∠ACB = 60° ⇒ ∠A = 30°

BC = 10 cm

P lies on AB and ∠PCB = 45°

Step 1:

In the big right triangle ABC (right at B) with ∠A = 30°, ∠C = 60°

For a 30 ⁣− ⁣60 ⁣− ⁣90 triangle, if the side opposite 30° is x, then hypotenuse = 2x and the other leg = `xsqrt3`

BC (opposite 30°) = 10 ⇒ AB = `10sqrt3`

In triangle BPC:

PB ⊥ BC and ∠PCB = 45°

ΔBPC is a 45 − 45 − 90 right triangle = BP = BC = 10 cm

AP = AB − BP

= `10sqrt3 − 10` cm or 7.32 cm