It is given that , =30^ , =50^ ,AB=5 cm,AC=8 cm and DF= 7.5 cm Th… - Vidyadip

MCQ

It is given that $\triangle\text{ABC}\sim\triangle\text{DFE},\angle\text{A}=30^\circ,\angle\text{C}=50^\circ ,\text{AB}=5 \text{ cm},\text{AC}=8\text{ cm}$ and $\text{DF}= 7.5\text{ cm}$ Then, the following is true :

A
$\text{DE}=12\text{ cm},\angle\text{F}=50^\circ$
✓
$\text{DE}=12\text{ cm},\angle\text{F}=100^\circ$
C
$\text{EF}=12\text{ cm},\angle\text{D}=100^\circ$
D
$\text{EF}=12\text{ cm},\angle\text{D}=30^\circ$

✓

Answer

Correct option: B.

$\text{DE}=12\text{ cm},\angle\text{F}=100^\circ$

$\triangle\text{ABC}\sim\triangle\text{DFE},\angle\text{A}=30^\circ,\angle\text{C}=50^\circ ,\text{AB}$
$=5 \text{ cm},\text{AC}=8\text{ cm}$ and $\text{DF}= 7.5\text{ cm}$
In triangle $\text{ABC},$
$\angle\text{a}+\angle\text{b}+\angle\text{c}=180^\circ$
$\angle\text{b}=180^\circ-30^\circ-50^\circ=100^\circ$
Since$\triangle\text{ABC}\sim\triangle\text{DFE},$ the corresponding angles are equal.
Thus,$\angle\text{D}=\angle\text{A}=30^\circ$
$\angle\text{F}=\angle\text{B}=100^\circ$
$\angle\text{E}=\angle\text{C}=50^\circ$
And
$\frac{\text{AB}}{\text{DF}}=\frac{\text{AC}}{\text{DE}}$
$\frac{5}{7.5}=\frac{8}{\text{DE}}$
$\text{DE}=\frac{(8\times7.5)}{5}=12\text{ cm}$

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