Question

AD is the bisector of BAC, if AB = 10 cm, AC = 6 cm and BC = 12 cm,find BD and DC.

Answer 1

Answer:

BD=7.5 cm and DC=4.5 cm

Step-by-step explanation:

Concept:

Angle bisector theorem:

when the vertical angle of a triangle is bisected, the bisector divides the base into two segments which have the same ratio as the order of other two sides

Let,BD = x

Then, DC=12-x

By angle bisector theorem,

$\frac{BD}{DC}=\frac{AB}{AC}\\\\\frac{x}{12-x}=\frac{10}{6}\\\\\frac{x}{12-x}=\frac{5}{3}$

3x=60-5x

3x+5x=60

8x=60

x=7.5

BD=7.5 cm and DC=4.5 cm

Answer 2

Answer:

BD = 7.5

DC = 4.5

Step-by-step explanation:

AD is the bisector of BAC, if AB = 10 cm, AC = 6 cm and BC = 12 cm,find BD and DC.

in Δ ADB    AB/Sin∠ADB  =  BD/Sin∠BAD   - eq 1

Similarly

in Δ ADC    AC/Sin∠ADC  =  DC/Sin∠CAD

∠BAD = ∠CAD = (1/2)∠BAC  (angle bisector)

∠ADC  + ∠ADB = 180°

∠ADC = 180° - ∠ADB

Sin∠ADC = Sin (180° - ∠ADB) = Sin∠ADB

Using these

AC/Sin∠ADB = DC/Sin∠BAD   - eq 2

Eq1 /Eq2

AB/AC  = BD/DC

=> 10/6 = BD /( 12 - BD)      (BD + DC = BC = 12) so DC = 12 -BD)

=>  120 - 10BD = 6BD

=> 120 = 16BD

BD = 120/16

=> BD = 7.5

 DC = 12 - 7.5 = 4.5