Question

If $\Delta ABC\sim \Delta PQR,\dfrac{ar\left( \Delta ABC \right)}{ar\left( \Delta PQR \right)}=\dfrac{9}{4}$ and $PQ=8$ , then find the length of AB.

Answer 1

Hint: To find the length of AB, we have to apply the area of similar triangles theorem which states that if two triangles are similar, then the ratio of the area of both triangles is proportional to the square of the ratio of their corresponding sides. We have to write this mathematically and solve it to get the required result.

Complete step by step answer:
We are given two similar triangles ABC and PQR whose areas are in the ratio 9:4. We have to find the measure of side AB.

We have to apply an area of similar triangles theorem. According to this theorem, if two triangles are similar, then the ratio of the area of both triangles is proportional to the square of the ratio of their corresponding sides. Therefore, we can write
$\dfrac{ar\left( \Delta ABC \right)}{ar\left( \Delta PQR \right)}=\dfrac{A{{B}^{2}}}{P{{Q}^{2}}}=\dfrac{B{{C}^{2}}}{Q{{R}^{2}}}=\dfrac{A{{C}^{2}}}{P{{R}^{2}}}=\dfrac{9}{4}$
Let us consider $\dfrac{A{{B}^{2}}}{P{{Q}^{2}}}=\dfrac{9}{4}$ . We are given $PQ=8$ . We have to substitute this value in the theorem.
$\Rightarrow \dfrac{A{{B}^{2}}}{{{8}^{2}}}=\dfrac{9}{4}$
Let us take the square root on both sides.
$\begin{align}
  & \Rightarrow \sqrt{\dfrac{A{{B}^{2}}}{{{8}^{2}}}}=\sqrt{\dfrac{9}{4}} \\
 & \Rightarrow \dfrac{AB}{8}=\dfrac{3}{2} \\
\end{align}$
We have moved the coefficient of AB to the RHS.
$\begin{align}
  & \Rightarrow AB=\dfrac{3}{2}\times 8 \\
 & \Rightarrow AB=\dfrac{3}{\require{cancel}\cancel{2}}\times {{\require{cancel}\cancel{8}}^{4}} \\
 & \Rightarrow AB=3\times 4 \\
 & \Rightarrow AB=12\text{ units} \\
\end{align}$
Hence, the length of side AB is 12 units.

Note: Students should learn the theorems and results associated with the congruent and similar triangles. They have a chance of making a mistake by stating the area of similar triangles theorem as the ratio of the area of two similar triangles is proportional to the ratio of their corresponding sides.