Question

A segment is cut from the circle of radius $14$ cm and the arc of segment makes ${90}^o$ angle at the center of the circle. Find the perimeter of the segment so cut.

• A. $2(11+7\sqrt{2})$ cm
• B. $(11+7\sqrt{2})$ cm
• C. $22+7\sqrt{2}$ cm
• D. $11+14\sqrt{2}$

Answer 1

The correct option is A $2(11+7\sqrt{2})$ cm

In order to find out the perimeter of segment we have to find out the length of arc$AB$ and the length of chord $AB$.

In $△ABC$, applying pythagoras theorem,
$A{B}^2=A{C}^2+B{C}^2$
$A{B}^2={14}^2+{14}^2$
$AB=\sqrt{2\times 196}$
$AB=14\sqrt{2}$ cm

Now, for arc,
Length of arc$AB=\frac{{90}^o}{{360}^o}\times 2\pi r$

arc$AB=\frac{1}{4}\times 2\times \frac{22}{7}\times 14$

arc$AB=22$ cm

So, $\text{Perimeter of segment}=\text{length of chord}AB+\text{length of arc}AB$
$\text{perimeter}=22+14\sqrt{2}$ cm
$\text{perimeter}=2(11+7\sqrt{2})$ cm

Hence, option $\left(a\right)$ correct.