Question

In a Δ ABC , AD is the bisector of ∠A, meeting side BC at D. 

(i) If BD = 2.5 cm, AB = 5 cm and AV = 4.2 cm, find DC. 

(ii) If BD = 2 cm, AB = 5 cm and DC = 3 cm, find AC. 

(iii) If AB = 3.5 cm, AC = 4.2 cm and DC = 2.8 cm, find BD. 

(iv) If AB = 10 cm, AC = 14 cm and BC = 6 cm, find BD and DC. 

(v) If AC = 4.2 cm, DC = 6 cm and BC = 10 cm, find AB. 

(vi) If AB = 5.6 cm, AC = 6 cm and DC = 6 cm, find BC. 

(vii) If AD = 5.6 cm, BC = 6 cm and BD = 3.2 cm, find AC. 

(viii) If AB = 10 cm, AC = 6 cm and BC = 12 cm, find BD and DC.

Answer 1

(i) we have 

Angle BAD=CAD 

Here AD bisects ∠A 

BD/DC=AB/AC 

2.5/DC=5/4.2 

DC=2.5 x 4.2/5 

DC=2.1 cm 

(ii) Here AD bisects ∠A 

AB/DC=AB/AC 

2/3=5/AC 

AC=15/2 

AC=7.5 cm 

(iii) in △ ABC A bisects ∠A 

BD/DC = AB/BC 

BD/2.8 = 3.5/4.2 

BD =3.5 x 2.8/4.2 

BD = 7/3 

BD = 2.33 cm 

(iv) In△ABC, AD bisects ∠A 

BD/DC = AB/AC 

X/6 - x = 10/14 

14x = 60 - 10x 

14x + 10x = 60 

24x = 60 

x = 60/24 

x = 5/2 

x = 2.5 

BD = 2.5 

DC = 6-2.5 

DC = 3.5 

(v) AB/AC=BD/DC

AB/4.2 = BC - DC/DC 

AB/4.2 = 10-6/6 

AB/4.2 = 4/6 

AB = 4 x 4.2/6 

AB = 2.8 cm 

(vi) BD/DC=AB/AC 

BD/6 = 5.6/6 

BD = 5.6

BC = BD + DC 

BC = 5.6 + 6 

BC = 11.6 cm 

(viii) In△ABC, AD bisects ∠A 

AB/AC = BD/DC 

5.6/AC = 3.2/BC - BD 

5.6/AC = 3.2/6 - 3.2 

5.6/AC = 3.2/2.8 

AC x 3.2 = 2.8 x 5.6 

AC = 2.8 x 5.6/3.2 

AC = 7 x 0.7 

AC = 4.9 cm 

(ix) let BD =x,then DC=12 - X 

BD/DC = AB/BC 

x/12-x = 10/6 

6x =120 - 10x 

6x + 10x = 120 

16x = 120 

x = 120/16 

x = 7.5 

BD = 7.5 cm 

DC = 12 - x 

DC =12 - 7.5 

DC = 4.5 cm