Question

Given a triangle ABC and D is a point on BC such that BD = 4 cm and DC = x cm. If ∠BAD = ∠C and AB = 8 cm, then, 

1. prove that triangle ABD is similar to triangle CBA. 
2. find the value of ‘x’.

Answer 1

Given:

Triangle ABC with D on BC, BD = 4 cm, DC = x cm, ∠BAD = ∠C, AB = 8 cm.

Show triangle ABD ~ triangle CBA and find x.

Step-wise calculation:

1. Similarity (AA):

∠BAD = ∠C   ...(Given)

∠ABD = ∠CBA   ...(They are the same angle at B)

Hence, ΔABD ~ ΔCBA by AA.

2. Corresponding sides from the similarity give

`(AB)/(BC) = (BD)/(AB)`

⇒ AB² = BD · BC

3. Substitute known lengths:

AB = 8

BD = 4

BC = BD + DC

BC = 4 + x

8² = 4(4 + x) 

64 = 16 + 4x

4x = 48

x = 12

DC = x = 12 cm.