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

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

Hence, Option 3 is the correct option.

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

1. OP = r

2. OP > r

3. OP ≥ r

4. OP < r

Answer

If P is a point on circle and O is center then OP = r but since, P is in the interior of the circle, so it will be smaller than the radius.

OP < r

Hence, Option 4 is the correct option.

The circumference of a circle must be

1. a positive real number

2. a whole number

3. a natural number

4. an integer

Answer

Circumference of circle = 2πr

Since, π = $\dfrac{22}{7}$ and radius is always a positive no.

∴ Circumference of a circle must be a positive real number.

Hence, Option 1 is the correct option.

AD is a diameter of a circle and AB is a chord. If AD = 34 cm and AB = 30 cm, then the distance of AB from the center of circle is

1. 17 cm

2. 15 cm

3. 4 cm

4. 8 cm

Answer

Let OM be the distance of AB from the center of the circle.

Since diameter = 34 cm, so radius = $\dfrac{34}{2}$ = 17 cm.

Since, the perpendicular to a chord from the centre of the circle bisects the chord,

∴ AM = MB = 15 cm.

In right triangle OAM,

⇒ OA² = AM² + OM² (By pythagoras theorem)

⇒ OM² = OA² - AM²

⇒ OM² = 17² - 15²

⇒ OM² = 289 - 225

⇒ OM² = 64

⇒ OM = $\sqrt{64}$ = 8 cm.

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. 12 cm

Answer

Let r be the radius of the circle.

From figure,

AC = AO + OC = r + r = 2r.

In right △ABC,

⇒ AC² = AB² + BC²

⇒ (2r)² = 12² + 16²

⇒ 4r² = 144 + 256

⇒ 4r² = 400

⇒ r² = $\dfrac{400}{4}$ = 100

⇒ r = $\sqrt{100}$ = 10 cm.

Hence, Option 3 is the correct option.

In the adjoining figure, O is the center of the circle. If OA = 5 cm, AB = 8 cm and OD ⊥ AB, then length of CD is equal to

1. 2 cm

2. 3 cm

3. 4 cm

4. 5 cm

Answer

Since, the perpendicular to a chord from the centre of the circle bisects the chord,

∴ AC = CB = $\dfrac{8}{2}$ = 4 cm.

In right △OAC,

⇒ OA² = OC² + AC²

⇒ 5² = OC² + 4²

⇒ OC² = 5² - 4²

⇒ OC² = 25 - 16

⇒ OC² = 9

⇒ OC = $\sqrt{9}$ = 3 cm.

From figure,

CD = OD - OC = 5 - 3 = 2 cm.

Hence, Option 1 is the correct option.

Consider the following two statements.

Statement 1: A line which meets a circle in two points is called a secant of the circle.

Statement 2: All radii of a circle are equal.

Which of the following is valid?

1. Both the statements are true.

2. Both the statements are false.

3. Statement 1 is true, and Statement 2 is false.

4. Statement 1 is false, and Statement 2 is true.

Answer

A secant is defined as a line that intersects a circle at exactly two distinct points.

∴ Statement 1 is true.

A radius is a line segment connecting the center of a circle to any point on its circumference.

By definition, all points on the circumference are equidistant from the center, meaning all radii have the same length.

∴ Statement 2 is true.

∴ Both the statements are true.

Hence, option 1 is correct option.