In Fig $9.21$ if $P R = 12 c m, Q R = 6 c m$ and $P L = 8 c m$ . then $Q M$ is

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Solution

$\begin{matrix} I n Δ P L R {(P R)}^{2} = {(P L)}^{2} + {(L R)}^{2} \Rightarrow {(12)}^{2} = 8^{2} + {(L R)}^{2} \Rightarrow L R = 4 \sqrt{5} c m n o w, L R = L Q + Q R \Rightarrow L Q = L R - Q R L Q = (4 \sqrt{5} - 6) C M A r e a o f Δ P L R = a r e a o f Δ P L Q + a r e a o f Δ P Q R \Rightarrow 16 \sqrt{5} = 16 \sqrt{5} - 24 + a r e a o f Δ P Q R \Rightarrow a r e a o f Δ P Q R = 24 c m^{2} a r e a o f Δ P L R = \frac{1}{2} \times L R \times P L \Rightarrow \frac{1}{2} \times 4 \sqrt{5} \times 8 \Rightarrow 16 \sqrt{5} c m^{2} a r e a o f Δ P L Q = \frac{1}{2} \times L Q \times P L \Rightarrow \frac{1}{2} (4 \sqrt{5} - 6) \times 8 \Rightarrow (16 \sqrt{5} - 24) c m^{2} \Rightarrow \frac{1}{2} \times P R \times Q M = 24 \Rightarrow \frac{1}{2} \times 12 \times Q M = 24 \Rightarrow Q m = 4 c m \end{matrix}$

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P Q = 10 c m, Q R = 8 c m, P R = 12 c m

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