MathematicsGiven a circle with the center indicated, which of the following is the value of x?

Given a circle with the center indicated, which of the following is the value of $x ?$

A
$100^{°}$
B
$80^{°}$
C
$50^{°}$
D
$40^{°}$

Solution:

Given figure,

From the figure,

∠ A = 50^{\circ}

.
The angle at the centre is twice the angle at the circumference subtended by the same arc.
Here, angle at the circumference is

x

and angle at the center

B

is

2 x

.
Angles opposite to equal sides are equal.
Here,

AB

and

BC

are the radius of the circle, they are of the same length.
So,

∠ A = ∠ C = 50^{\circ}

.
Now, in triangle ABC, by angle sum property,

\Rightarrow ∠ A + ∠ B + ∠ C = 180^{\circ}

\Rightarrow 2 x + 50^{°} + 50^{°} = 180^{°}

\Rightarrow 2 x + 10 0^{°} = 180^{°}

\Rightarrow 2 x = 180^{°} - 100^{°}

\Rightarrow 2 x = 80^{°}

\Rightarrow x = 40^{°}

Hence, the measure of

x = 40^{°}

.
Therefore, option 4 is correct.