Question

In the figure given below, $AB/\parallel EF\parallel CD$. If $AB=22.5cm,EP=7.5cm,PC=15cm$ and $DC=27$ cm. Find $AC$

• A. $AC=37.5$ cm
• B. $AC=22.5$ cm
• C. $AC=13.5$ cm
• D. $AC=27$ cm

Answer 1

The correct option is A $AC=37.5$ cm

Given: $AB\parallel EF\parallel CD$
$AB=22.5,EP=7.5,PC=15,DC=27$
In $△DCP$ and $△EPF$,
$\angle DCP=\angle PEF$ (Alternate angles)
$\angle CDP=\angle PFE$ (Alternate angles)
$\angle DPC=\angle EPF$ (Vertically opposite angles)
Thus, $△DCP\sim △FEP$ (AAA rule)
Now, $\frac{DC}{EF}=\frac{PC}{EP}$
$\frac{27}{EF}=\frac{15}{7.5}$
$EF=13.5$ cm
Now, In $△ABC$ and $△ECF$,
$\angle CEF=\angle CAB$ (Corresponding angles)
$\angle CFE=\angle CBA$ (Corresponding angles)
$\angle ECF=\angle ACB$ (Common angle)
Thus, $△ABC\sim △EFC$ (AAA rule)

Hence, $\frac{AB}{EF}=\frac{AC}{EC}$
$\frac{22.5}{13.5}=\frac{AC}{7.5+15}$
$AC=\frac{22.5\times 22.5}{13.5}$
$AC=37.5$ cm