Question

In Fig. 4.60, check whether AD is the bisector of ∠A of ∆ABC in each of the following:
(iii) AB = 8 cm, AC = 24 cm, BD = 6 cm and BC = 24 cm
(iv) AB = 6 cm, AC = 8 cm, BD = 1.5 cm and CD = 2 cm
(v) AB = 5 cm, AC = 12 cm, BD = 2.5 cm and BC = 9 cm.

Answer 1

SOLUTION :

(iii) GIVEN : AB = 8 cm, AC = 24 cm, BD = 6 cm and BC = 24 cm.

First check proportional ratio between sides.

DC = BC - BD

DC = 24 - 6

DC = 18

So,  AB/AC = BD/DC

8/24 = 6/18

⅓ = 1/3    

Here, AB/AC =  BD/CD

[If a line through one vertex of a triangle divides the opposite sides in the ratio of other two sides, then the line bisects the angle at the vertex.]

Hence, AD is the bisector of ∠ A.

(iv)  GIVEN : AB = 6 cm, AC = 8 cm, BD = 1.5 cm and CD = 2 cm.

First, check proportional ratio between sides.

So,  AB/AC = BD/DC

6/8 =1.5/2

¾ = 3/4    

Here, AB/AC =  BD/CD

[If a line through one vertex of a triangle divides the opposite sides in the ratio of other two sides, then the line bisects the angle at the vertex.]

Hence, AD is the bisector of ∠ A.

(v) GIVEN : AB = 5 cm, AC = 12 cm, BD = 2.5 cm and BC = 9 cm.

First, check proportional ratio between sides.

So,  AB/ AC = 5/12

BD/CD = 2.5/9 = 5/18

Here, AB/AC ≠ BD/CD

[If a line through one vertex of a triangle divides the opposite sides in the ratio of other two sides, then the line bisects the angle at the vertex.]

Hence, AD is not the bisector of ∠ A.

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