Question

In the given figure, $ST||RQ$, $PS=3$ cm and $SR=4$ cm. find the ratio of the area of $\Delta PST$ to the area of $\Delta PRQ$.

Answer 1

Hint: In this question we have been given with a figure which has two triangles in it. We have the triangles as $\Delta PST$ and $\Delta PRQ$. We also know the length of segments $PS$ and $SR$. We will solve this question by first proving $\Delta PST$ and $\Delta PRQ$ to be similar to each other. Once we prove them to be similar, we will use the property of the ratio of the area of similar triangles which is that the ratio of the area in similar triangles is given by the ratio of the squares of the corresponding side of the triangles.

Complete step by step answer:
We know that in the figure $ST||RQ$,$PS=3$ cm and $SR=4$ cm.
Now in triangles $\Delta PST$ and $\Delta PRQ$, we can see that:
$\Rightarrow \angle QPR=\angle TPS$ (common angle)
$\Rightarrow \angle PST=\angle PRQ$ (corresponding angles)
Since we have two angles equal, we can conclude that $\Delta PST\sim \Delta PRQ$ by angle-angle similarity.
Now we know the property of similar triangles that the ratio of the area in similar triangles is given by the ratio of the squares of the corresponding side of the triangles.
Consider the sides $PS$ and $PR$.
We know that $PS=3$, and from the figure we can see that $PR=PS+SR$ and we have $SR=4$.
Therefore, we get:
$\Rightarrow PR=3+4$
On simplifying, we get:
$\Rightarrow PR=7$
Now we have the ratio of the areas as:
$\Rightarrow \dfrac{Area\text{ }of\text{ }\Delta PST}{Area\text{ }of\text{ }\Delta PRQ}=\dfrac{{{3}^{2}}}{{{7}^{2}}}$
On squaring the terms, we get:
$\Rightarrow \dfrac{Area\text{ }of\text{ }\Delta PST}{Area\text{ }of\text{ }\Delta PRQ}=\dfrac{9}{49}$, which is the required solution.

Note: It is to be remembered that similarity of triangles is different from congruence of triangles. Congruent triangles are triangles which have the corresponding sides and the corresponding angles the same while as in similar triangles, some features of both the triangles are same which make them look similar.