Question

$\Delta DGH\text{and}\Delta DEF\text{are as shown below.}$ $\text{Find the lengths of segments DG and}\text{DE.}$
$\text{(All the measures are in cm.)}$

Answer 1

$\text{In}\Delta DGH\text{and}\Delta DEF$,

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

$\angle HDG=\angle FDE\text{(Common angle)}$

$\therefore \Delta DGH\sim \Delta DEF\text{(By AA similarity criterion)}$

$\Rightarrow \frac{DG}{DE}=\frac{DH}{DF}$

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

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

$\Rightarrow 36x-12=32x+16$

$\Rightarrow 36x-32x=16+12$

$\Rightarrow 4x=28$

$\Rightarrow x=\frac{28}{4}=7$

$\therefore DG=3x-1=3\times 7-1=21-1=20cm$

$\text{and}DE=4x+2=4\times 7+2=28+2=30cm$