Question

In the given figure, AB || EF || CD. If AB = 22.5 cm, EP = 7.5 cm, PC = 15 cm 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

Answer: (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, $\frac{AB}{EF}=\frac{AC}{EC}$

$\frac{22.5}{13.5}=\frac{AC}{7.5+15}$

$AC=\frac{22.5\ast 22.5}{13.5}$

$AC=37.5$ cm