Question

In the figure, m AOC = 48° in the circle centered at O. Find m ABC.

Answer 1

Answer: The measure of ∠ABC is 24°.

Step-by-step explanation: In the given figure, a circle is shown with centre at the point 'O'. And, m∠AOC = 48°.

We are to find the measure of ∠ABC.

In a CIRCLE, angles subtended by the same arc (or chord) at the circumference are equal in measures.

Also, angle subtended by an arc (or chord) at the centre of a circle is twice the angle subtended by the same arc (or chord) at the circumference.

In the attached figure, we draw AD such that the point 'D' lies on the circumference of the circle.

Now, since ∠AOC and ∠ADC are angles subtended by the arc AC at the centre and at the circumference respectively, so we have

m∠AOC = 2 × m∠ADC.

Therefore,

`m\angle ADC=(1)/(2)* m\angle AOC=(1)/(2)* 48^\circ=24^\circ.`

Again, ∠ADC and ∠ABC are angles subtended by the arc AC at the circumference of the circle.

So, m∠ABC = m∠ADC.

Since m∠ADC = 24°, therefore m∠ABC = 24°.

Thus, the measure of ∠ABC is 24°.

Answer 2

24°

Explanation:

Inscribed angle ABC subtending the same arc as central angle AOC has half the measure of AOC.

... 48°/2 = 24°