Question

In the given figure if $\angle GHE=\angle DFE={90}^{\circ }$, DH = 8 cm, DF = 12 cm, DG = 3x - 1 and DE = 4x + 2, then DG and DE are respectively ___.

• A. DG = 30 cm and DE = 20 cm
• B. DG = 10 cm and DE = 20 cm
• C. DG = 20 cm and DE = 30 cm
• D. DG = 20 cm and DE = 10 cm

Answer 1

The correct option is C

DG = 20 cm and DE = 30 cm

In $\Delta DFE\&\Delta DHG$

$\angle D=\angle D$ (common)

$\angle DFE=\angle DHG$ (given ${90}^{\circ }$)

So $\Delta DFE\sim \Delta DHG$ (AA similarity)

$\frac{DE}{DF}=\frac{DG}{DH}$ (corrosponding side are proportional)

$\Rightarrow \frac{4x+2}{12}=\frac{3x-1}{8}$

$\Rightarrow$ 8(4x + 2) = 12 (3x - 1)

$\Rightarrow$ 2(4x + 2) = 3 (3x - 1)

$\Rightarrow$ 8x + 4 = 9x - 3

$\Rightarrow$ x = 7 cm

So DG = 3x - 1 = 3 (7) - 1 = 20 cm

DE = 4x + 2 = 4(7) + 2 = 30 cm