Question

Find the area of rectangle

Answer 1

Given:

The rectangle and a quarter circle fitting towards one side of the rectangle and a semi circle which has a tangent as well.

To find:

Area of the rectangle?

Solution:

The given figure can be labeled as shown in the attached diagram.

AB and CD are the length, let $AB = CD = l$

AD and BC are the breadth, let $AD = BC = b$

The formula for area of a rectangle is:

$Area = Length \times Breadth$

As per the dimensions of the given rectangle:

$Area =l \times b$

Let the radius of the smaller semi circle be $r$.

Then we can clearly see that:

$l = b + r+r\\\Rightarrow l = b+2r \\OR\\\Rightarrow r = \dfrac{l-b}{2} .... (1)$

The $\triangle DOF$ is a right angled triangle as per the property of a tangent to a circle. And the right angle is at F.

Therefore, we can use Pythagorean Theorem in the $\triangle DOF$.

According to Pythagorean theorem:

$\text{Hypotenuse}^{2} = \text{Base}^{2} + \text{Perpendicular}^{2}\\\Rightarrow DO^{2} = DF^{2} + OF^{2}\\\Rightarrow (b+r)^2 = 5^2 +r^2\\\Rightarrow b^2+r^2+2br-r^2=25\\\Rightarrow b^2+2br=25\\\text{Using equation (1)}:\\\Rightarrow b^2+2b\dfrac{(l-b)}2=25\\\Rightarrow b^2+lb -b^2=25\\\Rightarrow \bold{l\times b = 25}$

Therefore, the area of the rectangle is $25\ cm^2$.