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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

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Solution

## (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 $⇒\frac{4}{6}=\frac{1.6}{2.4}\phantom{\rule{0ex}{0ex}}⇒\frac{2}{3}=\frac{2}{3}\phantom{\rule{0ex}{0ex}}$ (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 $⇒\frac{6}{8}=\frac{1.5}{2}\phantom{\rule{0ex}{0ex}}⇒\frac{3}{4}=\frac{3}{4}$ (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 .

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