Question

A circle has radius $\sqrt{2}cm$. It is divided into two segments by a chord of length $2cm$. Then the angle subtended by the chord at a point in major segment is:

• A. $75°$
• B. $60°$
• C. $80°$
• D. $45°$

Answer 1

The correct option is D

$45°$

Step 1: Given data

As per the data given in the question,

We have,

Radius of the circle $=\sqrt{2}cm$.

Length of the chord $=2cm$

Now drawing the figure,

From above-drawn figure,

$OA=OB=\sqrt{2}cm$

$AB=2cm$

Step 2-Applying the Pythagoras theorem:

Now, applying the Pythagoras theorem $△AOB$,

We will get,

$\begin{array}{rcl}O{A}^2+O{B}^2& =& A{B}^2\\ A{B}^2& =& {\left(\sqrt{2}\right)}^2+{\left(\sqrt{2}\right)}^2\\ A{B}^2& =& 2+2\\ A{B}^2& =& 4cm\\ 2^2& =& 4cm\end{array}$

As here, $O{A}^2+O{B}^2=A{B}^2$, So, we can say that the $△AOB$ will be a right-angled triangle.

So, $\angle AOB=90°$

Step 3 - Applying The angle subtended by an arc theorem of circle

As we know,

The angle subtended by an arc at the centre is twice the angle subtended at the circumference.

So, applying the above theorem,

We will get,

Value of the $\angle ACB$ will be half the value of the $\angle AOB$

So, value of $\angle ACB=\frac{\angle AOB}{2}=\frac{90°}{2}=45°$

Hence, option (D) will be the correct answer.