In the given figure, $∠ G H E = ∠ D F E = 90^{\circ}, D H = 8 c m, D F = 12 c m, D G = (3 x - 1) c m$ and $D E = (4 x + 2) c m$ . find the length of segments DG and DE.
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Solution

Theorem and Formulation: 2 Marks
Calculation: 2 Marks

$∠ G H D = ∠ D F E [E a c h 90^{\circ}] ∠ D = ∠ D [C o m m o n] ∴ Δ D H G \sim Δ D F E [A A S i m i l a r i t y] \frac{D H}{D F} = \frac{D G}{D E} \frac{8}{12} = \frac{3 x - 1}{4 x + 2} \Rightarrow 32 x + 16 = 36 x - 12 28 = 4 x \Rightarrow x = 7 ∴ D G = 3 x - 1 = 3 \times 7 - 1 = 21 - 20 c m D E = 4 x + 2 = 4 \times 7 + 2 = 28 + 2 = 30 c m$

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