Hushar Mulga
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∠ACB is inscribed in arc ACB of a circle with centre O. If ∠ACB = 65°, find m(arc ACB). 65° 130° 295° 230°

Chapter 3 – Circle – Text Book Solution

Problem Set 3 | Q 1.06 | Page 83

Four alternative answers for the following question is given. Choose the correct alternative. 

∠ACB is inscribed in arc ACB of a circle with centre O. If ∠ACB = 65°, find m(arc ACB).

  • 65°
  • 130°
  • 295°
  • 230°
solution

The measure of an inscribed angle is half of the measure of the arc intercepted by it.

By inscribed angle theorem,

∴ m∠ACB = `1/2` m(arc AB)

∴ m(arc AB) = 2m∠ACB

= 2 × 65º

= 130º

∴ m(arc ACB) = 360º − m(arc AB)

= 360º − 130º

= 230º ......[Measure of a circle is 360º]

Hence, the correct answer is 230°.

Explanation:_

An inscribed angle is an angle formed by two chords in a circle which have a common endpoint on the circle. The measure of an inscribed angle is half of the measure of the arc intercepted by it.

In this problem, we are given that m∠ACB is half of the measure of the arc AB, and we need to find the measure of the arc ACB.

Using the inscribed angle theorem, we know that:

m∠ACB = 1/2 m(arc AB)

We are given that m∠ACB = 65°. Substituting this value in the above equation, we get:

1/2 m(arc AB) = 65°

Simplifying this equation, we get:

m(arc AB) = 2 × 65° = 130°

Now, we need to find the measure of the arc ACB. We know that the sum of the measures of two arcs which are opposite to each other is 360°. Therefore:

m(arc ACB) = 360° – m(arc AB)

Substituting the value of m(arc AB), we get:

m(arc ACB) = 360° – 130° = 230°

Therefore, the measure of the arc ACB is 230°.

 

Chapter 3 – Circle – Text Book Solution

Problem Set 3 | Q 1.06 | Page 83

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