Question

In △ABC, AB = 3 and, AC = 4 cm and AD is the bisector of ∠A. Then, BD : DC is — (a) 9: 1 (b) 4:3 (c) 3:4 (d) 16:9
​

Answer 1

Correct Question -

In △ABC, AB = 3 cm and, AC = 4 cm .

AD is the bisector of ∠A.

Then, BD : DC is -

[ A ] 9 : 1

[ B ] 4 : 3

[ C ] 3 : 4

[ D ] 16 : 9

Solution-

Observe the given figure carefully .

Here ,

In ∆ABD and ∆ ACD

AD = AD

/_ BAD = /_ DAC

So, we can say that ∆ ABD and ∆ACD are similar .

Hence , the ratio of their corresponding sides must be equal .

Thus ,

AB / AC = BD / DC

=> 3 / 4 = BD / DC

Thus , BD : DC Is 3 : 4 .

Hence option C is the correct Option .

______________

Note -

This question can also be solved in a shorter method using the angle bisector theorem .

Answer 2

$\huge\sf\pink{Answer}$

☞ BD : DC = 3:4

$\rule{110}1$

$\huge\sf\blue{Given}$

✭ In ∆ABC,

$\:\:\:$◕ AB = 3 cm

$\:\:\:$◕ AC = 5 cm

$\:\:\:$◕ AD is a bisector of ∠A

$\rule{110}1$

$\huge\sf\gray{To \:Find}$

☆ BD : DC

$\rule{110}1$

$\huge\sf\purple{Steps}$

Here first let's prove that two triangles are similar,

In ∆ABD and ∆ACD

➝ AD = AD (Common)

➝ ∠BAD = ∠DAC (AD is the bisector of ∠A)

$\therefore$ ∆ABD ~ ∆ACD

We know that the Ratio of sides of corresponding triangles are equal.

So,

$\sf\dashrightarrow\dfrac{AB}{AC} = \dfrac{BD}{DC}$

$\sf\orange{\dashrightarrow\dfrac{3}{4} = \dfrac{BD}{DC}}$

$\rule{170}3$