Question

In the figure below, $\angle ABC=\angle DFE,\angle BAC=\angle FDE$, $D$ and $F$ are on $AB,AD=FB$ and the distances are shown in the figure itself in centimeters. Calculate the length of $AD$, in centimeters.

• A. $5$
• B. $4$
• C. $3$
• D. $1$
• E. $2$

Answer 1

Answer: (E)

In $△ABC$ and $△DFE,$

$\angle ABC=\angle DFE$ [Given]

$\angle BAC=\angle FDE$ [Given]

Hence, by $AAA$ rule of similarity,

$△ABC\sim △DFE$

As, the two triangles are similar their sides will be proportional as well,

$\therefore \frac{AB}{DF}=\frac{BC}{FE}=\frac{AC}{DE}\dots \left(i\right)$

Let $AD=BF=x$

$AB=AD+DF+BF$

$AB=6+2x$

From equation $\left(i\right),$

$\begin{array}{cc}\frac{AB}{DF}& =\frac{AC}{DE}\\ \frac{6+2x}{6}& =\frac{20}{12}\\ 72+24x& =120\\ 24x& =48\\ x& =2\end{array}$

Hence,

$AD=BF=2cm.$