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In a Δ ABC,...

Question

In a $Δ A B C$ , $D$ and $E$ are points on the sides $A B$ and $A C$ respectively such that $D E | | B C$ .
If $A D = 8 x - 7, D B = 5 x - 3, A E = 4 x - 3$ and $E C = (3 x - 1)$ , find the value of $x$ .

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Solution

$\begin{matrix} I n Δ A D E & Δ A B C ∠ A = ∠ A & ∠ A D E = ∠ A B C ∴ Δ A D E \sim Δ A B C [A A - s i m i l a r i t y] \frac{A D}{A B} = \frac{A E}{A C} \frac{8 x - 7}{8 x - 7 + 5 x - 3} = \frac{4 x - 3}{4 x - 3 + 3 x - 1} \Rightarrow (8 x - 7) (7 x - 4) = (4 x - 3) (13 x - 10) \Rightarrow 56 x^{2} - 32 x - 49 x + 28 = 52 x^{2} - 40 x - 39 x + 30 \Rightarrow 56 x^{2} - 81 x + 28 - 52 x^{2} + 79 x - 30 = 0 \Rightarrow 4 x^{2} - 2 x - 2 = 0 \Rightarrow 2 x^{2} - x - 1 = 0 \Rightarrow 2 x^{2} - 2 x + x - 1 = 0 \Rightarrow 2 x^{2} - 2 x + x - 1 = 0 \Rightarrow 2 x (x - 1) + 1 (x - 1) = 0 \Rightarrow (2 x + 1) (x - 1) = 0 ∴ 2 x + 1 = 0 a n d x - 1 = 0 ∴ x = \frac{- 1}{2} a n d x = 1 s i n c e l e n g t h c a n n o t b e - v e ∴ x = 1. \end{matrix}$

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