Question

The length and width of a rectangle are in ratio of 3:4. If the rectangle has an area of $300c{m}^2$, what is the length of the diagonal?
A. 15 cm
B. 12 cm
C. 25 m
D. 25 cm

Answer 1

Hint: We will first find the length and width of the rectangle by using the given ratio and area of the rectangle. Let the length of rectangle is $3x$ and the width of the rectangle is $4x$and then substitute these values in the formula is $l\times b$, where $l$ is the breadth of rectangle and $b$ is the width of the rectangle to find the value of $x$. Then, apply Pythagoras theorem in the triangle formed by the diagonal and sides of the rectangle to determine the length of the rectangle.

Complete step-by-step answer:
Let the common ratio of length and width of rectangle be $x$.
Then, the length of rectangle is $3x$ and the width of the rectangle is $4x$
We know that the area of the rectangle is $l\times b$, where $l$ is the breadth of the rectangle and $b$ is the width of the rectangle.
Then, the area of the given rectangle is $3x\left(4x\right)=12x^2$
But, we are given that the area of the rectangle is $300c{m}^2$.
$12x^2=300$
Divide both sides by 12
$$\begin{gathered} \Rightarrow x^2={\displaystyle \frac{300}{12}} \\ \Rightarrow x^2=25 \end{gathered}$$
Take square root on both sides,
$x=5$
Therefore, the length of the rectangle is $3\left(5\right)=15cm$ and the width of the rectangle is $4\left(5\right)=20cm$.
We have to find the length of the diagonal of the rectangle.

In a rectangle all angles are right angles.
Therefore, $ABC$ is a right triangle.
We want to find the length $AC$, which is the diagonal of the rectangle.
We will use Pythagoras theorem to find the value of $AC$.
Hence, ${\left(AC\right)}^2={\left(AB\right)}^2+{\left(BC\right)}^2$
On substituting the values of $AB$ and $BC$ in the above equation, we will get,
$$\begin{gathered} {\left(AC\right)}^2={\left(20\right)}^2+{\left(15\right)}^2 \\ \Rightarrow {\left(AC\right)}^2=400+225 \\ \Rightarrow {\left(AC\right)}^2=625 \end{gathered}$$
After taking square root on both sides, we will get,
$AC=25cm$
Also, the diagonals of the rectangle are equal.
Hence, the value of the rectangle is 25 cm
Thus, option D is correct.

Note: One should know the basic properties of a rectangle, such as all the angles of a rectangle are equal to ${90}^{\circ }$, opposite sides of a rectangle are equal and diagonals of a rectangle are equal.