Question

For a scale factor greater than 1, the ratio of the area of triangle to be constructed to the area of the given triangle will always be

• A. Less than 1
• B. Greater than 1
• C. Equal to 1
• D. Equal to 2

Answer 1

The correct option is B Greater than 1

Let the scale factor be m : n. Then, $\frac{m}{n}>1$.
$\Rightarrow m>n$
Let the triangle to be constructed be $\Delta ABC$ and the given triangle be $\Delta PQR$.
As $\Delta ABC\sim \Delta PQR$
$\Rightarrow \Delta ABC$ is a triangle whose sides are $\frac{m}{n}$ of the corresponding sides of $\Delta PQR$
Thus,$\frac{AB}{PQ}=\frac{BC}{QR}=\frac{CA}{RP}=\frac{m}{n}$.......(ii) (Ratio of corresponding sides of similar tringales are equal)

Now,$\frac{Area\left(\Delta ABC\right)}{Area\left(\Delta PQR\right)}={\left(\frac{AB}{PQ}\right)}^2$(Ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.)

$\Rightarrow \frac{Area\left(\Delta ABC\right)}{Area\left(\Delta PQR\right)}={\left(\frac{m}{n}\right)}^2$.......(iii) [Using (ii)]
Also, $m>n$ [From (i)]

Squaring both sides, we get
$m^2>n^2$
$\Rightarrow \frac{m}{n}>1$

$\Rightarrow (\frac{m}{n}>1)$

$\Rightarrow \frac{Area\left(\Delta ABC\right)}{Area\left(\Delta PQR\right)}>1$ [From (iii)]

Thus, the ratio of the area of triangle to be constructed $\left(\Delta ABC\right)$ to the area of triangle given $\left(\Delta PQR\right)$ is greater than 1.
Hence, the correct answer is option d.