If $L M ∥ A B, A L = x - 3, A C = 2 x,$ $B M = x - 2, B C = 2 x + 3.$ What is value of AC ?

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Solution

Given: LM ||AB
$A L = x - 3, A C = 2 x,$ $B M = x - 2, B C = 2 x + 3$

$\Rightarrow \frac{C L}{L A} = \frac{C M}{M B}$
(By Basic Proportionality Theorem)

Adding one on both sides, we get

$\frac{C L}{L A} + 1 = \frac{C M}{M B} + 1$

$\Rightarrow \frac{A C}{A L} = \frac{B C}{B M}$

$\Rightarrow \frac{2 x}{x - 3} = \frac{2 x + 3}{x - 2}$

$\Rightarrow 2 x^{2} - 4 x = 2 x^{2} - 6 x + 3 x - 9$

$\Rightarrow x = 9 u n i t s$

$\Rightarrow A C = 2 x = 2 \times 9 = 18 u n i t s$

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