Question : In ΔABC with sides 8 cm, 9 cm and 12 cm, the angle bisector of the largest angle divides the opposite sides into two segments. What is the length of the shorter segment?
Option 1: $5 \frac{11}{17} {~cm}$
Option 2: $4\frac{11}{17} {~cm}$
Option 3: $6\frac{13}{17} {~cm}$
Option 4: $3\frac{9}{17} {~cm}$
Correct Answer: $5 \frac{11}{17} {~cm}$
Solution :
Suppose AB = 8 cm, BC = 9 cm and AC = 12 cm
AD is the angle bisector of $\angle BAC$
We know that,
An angle bisector of a triangle divides the opposite side into two segments that are proportional to the other two sides of the triangle
⇒ $\frac{DC}{BD} = \frac{12}{8}$
⇒ $\frac{DC}{BD}+1 = \frac{3}{2}+1$
⇒ $\frac{DC + BD}{BD} = \frac52$
⇒ $\frac{9}{BD} = \frac52$
⇒ $BD = \frac{18}{5}$ cm
= $3\frac{3}{5}$
Hence, the correct answer is $3\frac{3}{5}$ cm.
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