ABC, which is right-angled at B, is inscribed in a circle with centre O and radius 6 units. If the length of the smaller arc between points A and B is 4units, what is the length of line segment BC?a)3b)c)33d)6e)2Correct answer is option 'D'. Can you explain this answer?

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ΔABC, which is right-angled at B, is inscribed in a circle with centre O and radius 6 units. If the length of the smaller arc between points A and B is 4π units, what is the length of line segment BC?

a)
3
b)
π
c)
3√3
d)
6
e)
2π

Correct answer is option 'D'. Can you explain this answer?

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ΔABC, which is right-angled at B, is inscribed in a circle with c...

Given:

The figure for the given information looks like this:

Since Triangle ABC is a right triangle inscribed in the circle, this means the hypotenuse AC of the triangle must be the diameter of the circle.
- Therefore, AC = 6*2 = 12 units
Let the length of BC be x units.
Length of the smaller arc between points A and B is 4π

units

To find: x = ?

Approach and Working:

We are given the length of the smaller arc between points A and B. Using this information, we can find ∠AOB
Finding ∠AOB

We’ll now use ∠AOB to find ∠BAC in isosceles triangle AOB

Now that we know ∠BAC, we can clearly infer that triangle ABC is 30-60-90 Triangle. Using the side-angle ratio property, we can find the value of x.
Finding x

Looking at the answer choices, we see that the correct answer is Option D.

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