If the arcs of the same lengths in two circles subtend angles 70° and 120° at the centre, determine the ratio of their radii?

Question

Question

This question was previously asked in

AAI ATC Junior Executive 25 March 2021 Official Paper (Shift 1)

Answer 1

Given:

Subtend angles are 70° and 120° at the center.

Formula Used:

Arc length (l) = r.θ

Where, r = radius

θ = central angle in radian

Calculation:

Here are two circles of different radii.

Let the radius of these two circles be 'r₁' and 'r₂'.

Length of the arc of the first circle (l) = r₁θ

⇒ \(l = r_1 \times 70° \times \frac{π}{180}\) [Multiply angle by π/180 to change into radian]

⇒ \(l = \frac{r_1 \times7 π}{18}\)

Length of the arc of the second circle (l) = r₂θ

⇒ \(l = r_2 \times 120° \times \frac{π}{180}\)

⇒ \(l = \frac{r_2\times 4π}{6}\)

Since the length of arcs are the same,

Hence, the Length of the first circle = the Length of the second circle.

⇒ = \(\frac{r_2\times 4π}{6}\)

⇒ \(\frac{r_1 }{r_2} = \frac{18 \times 4π}{6 \times 7π}\)

⇒ \(\frac{r_1}{r_2} = \frac{72}{42}\)

⇒ \(\frac{r_1}{r_2} = \frac{24}{14}\)

⇒ r₁ : r₂ = 24 : 14

∴ The ratio of radii of both the circles are 24 : 14.