Question

Which among the following is/are correct?
(I) If the altitudes of two similar triangles are in the ratio $2:1$, then the ratio of their areas is $4:1$.
(II) $PQ\parallel BC$ and $AP:PB=1:2$. Then, $\frac{A\left(△APQ\right)}{A\left(△ABC\right)}=\frac{1}{4}$

• A. $\left(I\right)$
• B. $\left(II\right)$
• C. Both $\left(I\right)$ and $\left(II\right)$
• D. None of the above

Answer 1

The correct option is A $\left(I\right)$

Option A: This statement is correct. The ratio of the altitudes of the similar triangles is $2:1$

Ratio of the areas of the similar triangles $=$ Square of the ratio of the altitudes.

$\therefore$ Ratio of the areas $={\left(\frac{2}{1}\right)}^2=4:1$

Option B: If $PQ\parallel BC$, then $△APQ\sim △ABC$ by AA test of similarity.

Hence, $\frac{A\left(△APQ\right)}{A\left(△ABC\right)}=\frac{A{P}^2}{A{B}^2}$

If $AP=x$ and $BP=2x$, then $AB=3x$.

$\therefore \frac{A\left(△APQ\right)}{A\left(△ABC\right)}=\frac{1}{9}$

So, the given statement is false