Question

If the circumference of the below circle is $4\pi$ units, then the length of $AB$ is

• A. $\sqrt{2}$
• B. $4$
• C. $4\sqrt{2}$
• D. $2\sqrt{2}$

Answer 1

The correct option is D $2\sqrt{2}$


Given, the circumference of the circle is $4\pi$ units.

The circumference of the circle is given by $C=2\pi r,$ where $r$ is the radius of the circle.
$\therefore 4\pi =2\pi \times r$
Divide both sides by $2\pi$
$\Rightarrow \frac{4\pi }{2\pi }=\frac{2\pi \times r}{2\pi }$
$\Rightarrow \frac{2\times 2\pi }{2\pi }=r$
$\Rightarrow 2=r$
$\Rightarrow r=2$ units

Since $OA$ and $OB$ are the distances from the center of the circle to the circumference, they represent the radius of the circle.
$\therefore OA=OB=2$

Using pythagorean theorem in $△OAB,$ we get
$A{B}^2=O{A}^2+O{B}^2$
$\Rightarrow A{B}^2=2^2+2^2\Rightarrow A{B}^2=4+4$
$\Rightarrow A{B}^2=8$
Taking square root both sides
$\Rightarrow \sqrt{A{B}^2}=\sqrt{8}$
$\Rightarrow AB=\sqrt{2\times 2\times 2}$
$\Rightarrow AB=\pm 2\sqrt{2}$
$\Rightarrow AB=2\sqrt{2}(\because \text{A side of a triangle cannot be negative}$
$\therefore AB=-2\sqrt{2}\text{is rejected})$

Hence, option (d.) is the correct choice.