If the length of an arc of a circle of radius r is equal to that of an arc of a circle of radius 2r, then the angle of the corresponding sector of the first circle is double the angle of the corresponding sector of the other circle. Is this statement false? Why?

#### Options

True

False

#### Solution

This statement is **False.**

**Explanation:**

Let two circles C_{1} and C_{2} of radius r and 2r with centres O and O’, respectively.

It is given that, the arc length `hat(AB)` of C_{1} is equal to arc length `hat(CD)` of C_{2} i.e., `bar(AB) = bar(CD) = l` ....(say)

Now, let θ_{1} be the angle subtended by arc `bar(AB)` of θ_{2} be the angle subtended by arc `bar(CD)` at the centre.

∴ `bar(AB) = l = Q_1/360 xx 2pir` ......(i)

And `hat(CD) = l = theta_2/360 xx 2pi (2r) = theta_2/360 xx 4pil` .....(ii)

From equations (i) and (ii),

`theta_1/360 xx 2pir = theta_2/360 xx 4pir`

⇒ `theta_1 = 2theta_2`

i.e., Angle of the corresponding sector of C_{1} is double the angle of the corresponding sector of C_{2}.

It is true statement.