First slide

Properties of triangles

Question

If, in a $△ A B C, (a + b + c) (b + c - a) = λ b c,$ then

Moderate

A

$λ < 0$
B

$λ > 4$
C

$λ > 0$
D

$0 < λ < 4$

Solution

We have,
$\begin{array}{l} (a + b + c) (b + c - a) = λ b c \\ \Rightarrow 2 s (2 s - a) = λ b c \\ \Rightarrow \frac{s (s - a)}{b c} = \frac{λ}{4} \\ \Rightarrow \cos^{2} \frac{A}{2} = \frac{λ}{4} \\ \Rightarrow 0 < \frac{λ}{4} < 1 \Rightarrow 0 < λ < 4 \\ \Rightarrow [∵ \cos^{2} \frac{A}{2} \leq 1] [∵ \cos^{2} \frac{A}{2} \leq 1] \end{array}$

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