Chapter 12, Problem 12CT

To determine

To calculate: The area of a square which diagonal is $24\text{cm}$ long and when diagonal of square is $16\text{cm}$ then area of square is $128{\text{cm}}^2$ .the area of square is directly proportional to the square of its diagonal.

Answer to Problem 12CT

The area of square when diagonal of square $24\text{cm}$ is $288{\text{cm}}^2$ .

Explanation of Solution

Given information:

The area of square is directly proportional to the square of its diagonal and diagonal of square is $16\text{cm}$ then area of square is $128{\text{cm}}^2$ .

Formula used:

The variable $y$ is directly proportional to $x^2$ as,

  $y=k{x}^2$ , where $k$ is a nonzero constant this equation is known as quadratic direct variation.

Calculation:

Let side of the square is $a$ and diagonal is $b$ .the area of square is ${\left(\text{side}\right)}^2$ so by area formula area of square is ${\left(a\right)}^2$ .

From the provided line the area of square is directly proportional to the square of its diagonal,

  ${\left(a\right)}^2=k{b}^2$

Since, the area of square is $128{\text{cm}}^2$ when diagonal of that square is $16\text{cm}$ .put the values in equation,

  $\begin{array}{c}128=k{\left(16\right)}^2\\ k=\frac{128}{256}\\ k=\frac{1}{2}\end{array}$

The value of $k=\frac{1}{2}$ .now if diagonal of square is $24\text{cm}$ then area is,

  $\begin{array}{l}{\left(a\right)}^2=\frac{1}{2}{\left(24\right)}^2\\ {\left(a\right)}^2=\frac{576}{2}\\ {\left(a\right)}^2=288\end{array}$

So, the area of square when diagonal of square is $24\text{cm}$ is $288{\text{cm}}^2$