ANGLE BISECTOR THEOREM EXAMPLE PROBLEMS

Angle bisector theorem :

The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the corresponding sides containing the angle.

In ΔABC, AD is the internal bisector

AB/AC  =  BD/CD

Example 1 :

In a triangle ABC,AD is the internal bisector of angle A, meeting BC at D.

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

Solution :

Using “Angle bisector theorem” in the triangle ABC, we get

(AB/AC)  = ( BD/DC)

(5/AC)  =  (2/3)

AC  =  (3  5)/2

      =  15/2

      =  7.5 cm  

(ii) If AB = 5.6 cm, AC = 6 cm and DC = 3 cm find BC.

Solution :

Using “Angle bisector theorem” in the triangle ABC, we get

(AB/AC)  =  (BD/DC)

(5.6/6)  =  (BD/3)

BD  =  (5.6 ⋅ 3)/6

  =  16.8/6

  =  2.8 cm

From this we need to find the value of BC

 BC  =  BD + DC

  =  2.8 + 3

.    =  5.8 cm  

(iii) If AB = x, AC = x – 2 cm, BD = x + 2 cm and DC = x – 1 find the value of x

Solution :

Using “Angle bisector theorem” in the triangle ABC, we get

(AB/AC)  =  (BD/DC)

[x/(x – 2)]  =  [(x + 2)/(x - 1)]

x (x – 1)  =  (x + 2) (x – 2)

x ² – x  =  x ² – 4

x² – x - x² + 4  =  0

   - x + 4  =  0

   - x  =  - 4

     x  =  4 cm

The value of x is 4 cm

Verifying Angle Bisector Theorem in Given Triangle

Example 2 :

Check whether AD is the bisector of angle A of the triangle ABC in each of the following.

(i) AB = 4 cm, AC = 6 cm, BD = 1.6 cm and CD = 2.4 cm

Solution :

To check whether AD is the bisector of angle A of the triangle ABC, we have to check the following condition

(AB/AC)  =  (BD/DC)

(4/6)  =  (1.6/2.4)

0.66  =  0.66

From this we come to know that AD is the bisector of angle A of the triangle ABC.

(ii) AB = 6 cm, AC = 8 cm, BD = 1.5 cm and CD = 3 cm

Solution :

To check whether AD is the bisector of angle A of the triangle ABC, we have to check the following condition

(AB/AC)  =  (BD/DC)

(6/8)  =  (1.5/3)

0.75  ≠  0.5

Hence, AD is not the bisector of angle A of the triangle ABC.

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