If A, B, C are three points on a circle with center O such that ∠AOB=90° and ∠BOC=120°, then what will be the measure of ∠AOC?

A
$60 °$
B
$75 °$
C
$90 °$
D
$135 °$

Solution:

Given, A, B, C are three points on a circle with center O such that

∠ AOB = 90 °

and

∠ BOC = 120 °

.
Construct a figure according to the given data,

Here,

∠ AOC = 360 ° - (120 ° + 90 °) = 150 °

But ∠AOC is at the center made by are AC and ∠ABC at the remaining part of the circle.
Angle subtended by the arc at the center is double the angle subtended by it at any point on the circle.

∠ ABC = \frac{1}{2} ∠ AOC

= \frac{1}{2} \times 150 °

= 75 °

Hence, the measure of ∠AOC is

75 °

.
Therefore, option 2 is correct.

If A, B, C are three points on a circle with center O such that $∠ AOB = 90 °$ and $∠ BOC = 120 °$ , then what will be the measure of $∠ AOC$ ?

Solution:

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