Circle — Multiple Choice Questions

If P and Q are any two points on a circle, then the line segment PQ is called a

1. radius of the circle

2. diameter of the circle

3. chord of the circle

4. secant of the circle

Answer

We know that,

A chord of a circle is a line segment with its end points on the circle.

Here, P and Q are two points lying on the circle.

∴ The line segment PQ is a chord of the circle.

Hence, option 3 is the correct option.

If P is a point in the interior of a circle with centre O and radius r, then

1. OP = r

2. OP > r

3. OP ≥ r

4. OP < r

Answer

We know that,

A point lies in the interior of a circle if its distance from the centre is less than the radius of the circle.

Here, P lies in the interior of the circle with centre O and radius r.

∴ OP < r

Hence, option 4 is the correct option.

If AB = 12 cm, BC = 16 cm and AB is perpendicular to BC, then the radius of the circle passing through the points A, B and C is

1. 6 cm

2. 8 cm

3. 10 cm

4. 20 cm

Answer

Given, AB = 12 cm, BC = 16 cm and AB ⊥ BC.

Since, ∠ABC = 90°, the chord AC subtends a right angle at the point B of the circle.

So, AC is the diameter of the circle passing through the points A and C.

In ∆ABC, using Pythagoras theorem,

⇒ AC² = AB² + BC²

⇒ AC² = 12² + 16²

⇒ AC² = 144 + 256

⇒ AC² = 400

⇒ AC = $\sqrt{400}$

⇒ AC = 20 cm

As we know that radius is half of the diameter.

⇒ Radius = $\dfrac{1}{2} \times 20$

⇒ Radius = 10 cm

Hence, option 3 is the correct option.

In the adjoining figure, AB is a diameter of the circle. If AC = BC, then ∠CAB is equal to

1. 30°

2. 45°

3. 60°

4. 90°

Answer

Given, AB is a diameter of the circle and AC = BC.

⇒ ∠CAB = ∠CBA [Angles opposite to equal sides of a triangle are always equal]

We know that,

Angle in a semicircle is a right angle.

⇒ ∠ACB = 90°

In ∆ACB, using angle sum property,

⇒ ∠CAB + ∠CBA + ∠ACB = 180°

⇒ ∠CAB + ∠CAB + 90° = 180°

⇒ 2∠CAB = 180° - 90°

⇒ 2∠CAB = 90°

⇒ ∠CAB = $\dfrac{90\degree}{2}$

⇒ ∠CAB = 45°

Hence, option 2 is the correct option.