Circle — Statement I-II Type Questions

Statement I: A chord of a circle passing through its centre is called a diameter.

Statement II: Diameter = 2 × radius.

1. Statement I is true but Statement II is false.

2. Statement I is false but Statement II is true.

3. Both Statement I and Statement II are true.

4. Both Statement I and Statement II are false.

Answer

A chord of a circle which passes through the centre of the circle is called a diameter of the circle.

So, statement I is true.

A diameter of a circle is made up of two radii lying on the same straight line, so its length is twice the radius of the circle.

⇒ Diameter = 2 × radius

So, statement II is true.

∴ Both statement I and statement II are true.

Hence, option 3 is the correct option.

Statement I: In the adjoining figure, AB is the diameter. If the radius of the circle is 5 cm and PB = 6 cm, then AP = 7 cm.

Statement II: The arc $\overset{\frown}{APB}$ = the arc $\overset{\frown}{AQB}$.

1. Statement I is true but Statement II is false.

2. Statement I is false but Statement II is true.

3. Both Statement I and Statement II are true.

4. Both Statement I and Statement II are false.

Answer

Given, the radius of the circle = 5 cm and PB = 6 cm.

As we know that the diameter is twice the radius.

⇒ AB = 2 × 5

⇒ AB = 10 cm

We know that,

Angle in a semicircle is a right angle.

⇒ ∠APB = 90°

In ∆APB, using Pythagoras theorem,

⇒ AP² + PB² = AB²

⇒ AP² + 6² = 10²

⇒ AP² + 36 = 100

⇒ AP² = 100 - 36

⇒ AP² = 64

⇒ AP = $\sqrt{64}$

⇒ AP = 8 cm

So, statement I is false.

Since AB is the diameter of the circle, it divides the circle into two equal arcs and each of these arcs is a semicircle.

⇒ arc APB = arc AQB

So, statement II is true.

∴ Statement I is false but statement II is true.

Hence, option 2 is the correct option.