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Equal Intercept Theorem

In the figure...

Question

In the figure if $D E ∥ B C$ and $A D = 3 x - 2$ , $A E = 5 x - 4$ , $B D = 7 X - 4$ , $B D - 7 x - 5$ AND $C E = 5 x - 3$ . Find x.

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Solution

Given : In $△ A B C, D E ∥ B C$ we have to find the value of x
$△ A B C, D E ∥ B C$ We have
$∴ \frac{A D}{B D} = \frac{A E}{C E}$ (Using Thales Theorem)

$\Rightarrow \frac{3 x - 2}{7 x - 5} = \frac{5 x - 4}{5 x - 3}$

$\Rightarrow (3 x - 2) (5 x - 3) = (5 x - 4) (7 x - 5)$
$\Rightarrow 15 x^{2} - 10 x - 9 x + 6 = 35 x^{2} - 25 x - 28 x + 20$
On solving we get,
$\Rightarrow 10 x^{2} - 17 x + 7 = 0$
$\Rightarrow 10 x (x - 1) - 7 (x - 1) = 0$
$\Rightarrow (x - 1) (10 x - 7) = 0$
x=1 or x= $\frac{7}{10}$
Hence, $x = 1$ or $\frac{7}{10}$

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Q. In the figure shown

D E ∥ B C

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