Given: $∠ G H E = ∠ D F E = 90^{\circ}$ (IMAGE)
$D H = 8, D F = 12, D G = 3 x - 1$ and $D E = 4 x + 2.$ Find the
lengths of segments $D G$ and $D E$ .

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Solution

Given: $∠ G H E = ∠ D F E = 90^{o}$
$D H = 8^{'} D F = 12$
$D G = 3 x - 1, D E = 4 x = 2$
In $Δ G H D$ & $Δ E F D$
$∠ D = ∠ D$ (common)
$∠ G H D = ∠ D E F = 90^{o}$ (Given)
$∴ G H D \sim Δ E F D$
$\frac{D F}{D H} = \frac{D E}{D G}$
$\frac{12}{8} = \frac{4 x = 2}{3 x - 1}$
$9 x - 3 = 8 x + 4$
$x = 7$
$D G = 3 x - 1 = 21 - 1 = 20$
$D E = 4 x + 2 = 28 + 2 = 30$

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