Question

$AB,EF$ and $CD$ are parallel lines. Given that $EG=5$ cm, $GC=10$ cm, $AB=15$ cm and $DC=18$ cm. What is the value of $AC?$

• A. $20$ cm
• B. $24$ cm
• C. $25$ cm
• D. $28$ cm

Answer 1

The correct option is C $25$ cm

Given, $AB\parallel EF\parallel CD$
In $△EFG$ and $△CDG,$
$\angle EFG=\angle GDC$ ($EF\parallel CD$, alt. $\angle$s are equal)
$\angle EGF=\angle CGD$ (vert. opp. $\angle$s)
$\therefore$ $△EFG\sim △GCD$ (By AA similarity)
$\therefore$ $\frac{EG}{GC}=\frac{EF}{DC}\Rightarrow \frac{EF}{18}=\frac{5}{10}\Rightarrow EF=9$ cm
Now in $△$s $ABC$ and $EFC,$
$\angle ACB=\angle ECF$ (common)
$\angle ABC=\angle EFC$ ($AB\parallel EF,$ corr. $\angle$s are equal)
$\therefore$ $△ABC\sim △EFC$ (By AA similarity)
$\Rightarrow$ $\frac{AC}{EC}=\frac{AB}{EF}\Rightarrow \frac{AC}{(EG+GC)}=\frac{AB}{EF}$
$\Rightarrow$ $\frac{AC}{(5+10)}=\frac{15}{9}\Rightarrow AC=25$ cm.