Question : Points $P$ and $Q$ lie on sides $AB$ and $AC$ of triangle $ABC$, respectively, such that segment $PQ$ is parallel to side $BC$. If the ratio of areas of triangle $APQ$ to triangle $ABC$ is 25 : 36, then the ratio of $AP$ to $PB$ is:
Option 1: $5:6$
Option 2: $1:5$
Option 3: $6:5$
Option 4: $5:1$
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Correct Answer: $5:1$
Solution : We have: $\frac{\text{area of } \Delta APQ}{\text{area of }\Delta ABC}=\frac{25}{36}$ Two triangles $APQ$ and $ABC$ such that $△APQ ∼△ABC$ $⇒\frac{\text{area of }\Delta APQ}{\text{area of }\Delta ABC}=(\frac{AP}{AB})^2$ $⇒\frac{25}{36}=(\frac{AP}{AB})^2$ $⇒\frac{5}{6}=(\frac{AP}{AB})$ If $AB$ = 6 units and $AP$ = 5 units, then $PB = AB - AP$ = 1 unit So, the ratio of $AP:PB$ = $5:1$ Hence, the correct answer is $5:1$.
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