Question

In ΔABC, BD ⊥ AC. ∠A = 45°, ∠C = 30° and BC = 20 cm. Find the length of (i) BD and (ii) AB.

Answer 1

Given:

∠A = 45°, ∠C = 30° and BC = 20 cm

BD ⊥ AC

Finding BD & AB

Step 1: Find ∠B

= ∠B = 180° − (∠A + ∠C) 

= 180° − (45° + 30°)

= 105°

Step 2: Use sine rule in ΔABC

= `(AB)/(sin C) = (BC)/(sin A)`

= AB = `(BC ⋅ sin C)/sin A`

= `(20 ⋅ sin 30°)/(sin 45°)`

= `(20 ⋅ 1/2)/(1/sqrt2)`

= `10/(1/sqrt2)`

= `10sqrt2`

so AB = `10sqrt2` cm

Step 3:

Since ∠A = 45°

= BD = AB ⋅ sin 45°

= `10sqrt2 ⋅ 1/sqrt2 = 10`

= BD = 10 cm