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The Mid-Point Theorem

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 $A D = 4, A E = 8, D B = x - 4$ , and $E C = 3 x - 19$ , find $x$ .

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Solution

$\begin{matrix} I n Δ A B C & Δ A D E ∠ A D E = ∠ A B C ∠ A = ∠ A ∴ Δ A B C \sim Δ A D E (A A - S i m i l a r i t y) ∴ \frac{A B}{A D} = \frac{A C}{A E} \Rightarrow \frac{A B}{A D} - 1 = \frac{A C}{A E} - 1. \Rightarrow \frac{A B - A D}{A D} = \frac{A C - A E}{A E} \Rightarrow \frac{D B}{A D} = \frac{E C}{A E} \Rightarrow \frac{x - 4}{4} = \frac{3 x - 19}{8} \Rightarrow 8 x - 32 = 12 x - 76 \Rightarrow 4 x = 76 - 32 x = \frac{44}{4} = 11 \end{matrix}$

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