It is given that . If =30^ , =50^ ,AB=5 cm,AC=8 cm and DF=7.5 cm then which of the following is true?

It is given that . If =30^ , =50^ ,AB=5 cm,AC=8 cm and DF=7.5 cm …

MCQ

It is given that $\triangle\text{ABC}\sim\triangle\text{DFE}.$ If $\angle\text{A}=30^\circ,\angle\text{C}=50^\circ,\text{AB}=5\text{ cm},\text{AC}=8\text{ cm}$ and $\text{DF}=7.5\text{ cm}$ then which of 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.

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

Given that,
$\angle\text{A}=30^\circ,\angle\text{C}=50^\circ$
$\triangle\text{ABC}\sim\triangle\text{DFE}$
$\Rightarrow\angle\text{A}=\angle\text{D}=30^\circ$
$\angle\text{C}=\angle\text{E}=50^\circ$
Using angle sum property,
we can find $\angle\text{B}=100^\circ$
So, $\angle\text{B}=\angle\text{F}=100^\circ$
Also $, AB = 5\ cm, AC = 8\ cm$ and $DF = 7.5\ cm$
$\frac{\text{AB}}{\text{DF}}=\frac{\text{BC}}{\text{FE}}=\frac{\text{AC}}{\text{DE}}$
$\Rightarrow\frac{5}{7.5}=\frac{\text{BC}}{\text{FE}}=\frac{8}{\text{DE}}$
$\Rightarrow\frac{5}{7.5}=\frac{8}{\text{DE}}$
$\Rightarrow\frac{8\times7.5}{5}=12\text{ cm}$
Hence $, DE = 12\ cm$ and $\angle\text{F}=100^\circ$