In a Δ A B C , if 4 a = b + c , then tan B 2 tan C 2 =

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Question: Answered & Verified by Expert

In a

Δ A B C,

if

4 a = b + c,

then

\tan \frac{B}{2} \tan \frac{C}{2} =

MathematicsProperties of TrianglesTS EAMCETTS EAMCET 2021 (05 Aug Shift 1)

Options:

A $\frac{1}{3}$
B $\frac{3}{5}$
C $\frac{2}{3}$
D $\frac{1}{2}$

Solution:

1379 Upvotes Verified Answer

The correct answer is:

\frac{3}{5}

$\tan \frac{B}{2} \tan \frac{C}{2}$

$= \sqrt{\frac{(s - a) (s - c)}{s (s - b)}} \cdot \sqrt{\frac{(s - a) (s - b)}{s (s - c)}}$

$= \sqrt{\frac{{(s - a)}^{2}}{s^{2}}} = \frac{s - a}{s}$

$= \frac{\frac{a + b + c}{2} - a}{\frac{a + b + c}{2}}$ ,

Given, $b + c = 4 a$

$= \frac{\frac{a + 4 a}{2} - a}{\frac{a + 4 a}{2}}$

$= \frac{3}{5}$

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