Question

In Fig. 4.60, check whether AD is the bisector of ∠A of ∆ABC in each of the following:

(i) AB = 5 cm, AC = 10 cm, BD = 1.5 cm and CD = 3.5 cm
(ii) AB = 4 cm, AC = 6 cm, BD = 1.6 cm and CD = 2.4 cm
(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

(i) It is given that,, and .

We have to check whether is bisector of .

First we will check proportional ratio between sides.

Now

$\frac{AB}{AC}=\frac{5}{10}=\frac{1}{2}$

$\frac{BD}{CD}=\frac{1.5}{3.5}=\frac{3}{7}$

Since $\frac{AB}{AC}\ne \frac{BD}{CD}$

Hence is not the bisector of .

(ii) It is given that,,, and .

We have to check whether is bisector of .

First we will check proportional ratio between sides.

So
$$\begin{gathered} \Rightarrow \frac{4}{6}=\frac{1.6}{2.4} \\ \Rightarrow \frac{2}{3}=\frac{2}{3} \end{gathered}$$

(It is proportional)

Hence, is bisector of .

(iii) It is given that,,, and .

We have to check whether is bisector of .

First we will check proportional ratio between sides.

Now

So

(It is proportional)

Hence, is bisector of .

(iv) It is given that, AB = 6 cm, AC = 8 cm, BD = 1.5 cm and CD = 2 cm.

We have to check whether is bisector of .

First we will check proportional ratio between sides.

So

$$\begin{gathered} \Rightarrow \frac{6}{8}=\frac{1.5}{2} \\ \Rightarrow \frac{3}{4}=\frac{3}{4} \end{gathered}$$

(It is proportional)

Hence is bisector of .

(v) It is given that AB = 5 cm, AC = 12 cm, BD = 2.5 cm and BC = 9 cm

We have to check whether is bisector of .

First we will check proportional ratio between sides.

Now

$\frac{AB}{AC}=\frac{5}{12}$

$\frac{BD}{CD}=\frac{2.5}{9}=\frac{5}{18}$

Since $\frac{AB}{AC}\ne \frac{BD}{CD}$

Hence is not the bisector of .