Question

A square has side length of $16.5\mathrm{in}$

If the area is multiplied by $9$, what happens to the side length?

• A. The side length is doubled.
• B. The side length is multiplied by 4.5
• C. The side length is multiplied by 9
• D. The side length is tripled.

Answer 1

The correct option is D

The side length is tripled.

Find the relationship between multiple of sides of square and multiple of area of square:

The explanation for the correct option:

Given that the area of a square is $16.5\mathrm{in}.$

Let $\mathrm{a}$ be the side of the triangle.

Then

$a^2=16.5\mathrm{in}.$

Multiply $9$ to the area of the square i.e. both side of the above equation:

$$\begin{gathered} 9\times {\mathrm{a}}^2=9\times 16.5\mathrm{in}. \\ {\left(3\mathrm{a}\right)}^2=9\times 16.5\mathrm{in}. \end{gathered}$$

Thus in order to area get multiplied to $3$, the side length is tripled.

Thus the correct option is option $D$.

Explanation for incorrect option:

Option $A$. If the side length is double the are become as follow

$\begin{array}{rcl}\mathrm{Area}\text{'}& =& {\left(2\mathrm{a}\right)}^2\\ & =& 4{\mathrm{a}}^2\\ & =& 4\times 16.5\mathrm{in}.\end{array}$

If the side length is double they are get multiplied to $4$

Hence, Option $A$ is the incorrect option.

Option $B$. If the side length is multiplied by $4.5$ the area become as follows:

$\begin{array}{rcl}\mathrm{Area}\text{'}& =& {\left(4.5\mathrm{a}\right)}^2\\ & =& 20.25{\mathrm{a}}^2\\ & =& 20.25\times 16.5\mathrm{in}.\end{array}$

If the side length is double they are get multiplied to $4.5$, the area become $20.25$ times

Hence, Option $B$ is the incorrect option.

Option $C$. If the side length is multiplied by $4.5$ the area becomes as follows:

$\begin{array}{rcl}\mathrm{Area}\text{'}& =& {\left(9\mathrm{a}\right)}^2\\ & =& 81{\mathrm{a}}^2\\ & =& 81\times 16.5\mathrm{in}.\end{array}$

If the side length is double they are get multiplied to $9$, the area becomes $81$ times

Hence, Option $C$ is the incorrect option.

Hence, Option $D$ is the correct option.