Question

in figure two congruent circles have centres O and O'.Arc AXBsubtends an angle of 75degree at the centre O and arc AYB Subtends an angle of 25degree at the centre O'. Then the ratio of arcs AXB and AYB is

Answer 1

3 : 1

Congruent circles

They are defined as circles with radii that are the same or equal.

Given that the circles are congruent,

Therefore, their radius will be the same.

Length of arc AXB = ($\frac{\theta}{360} * 2\pi r$)

Length of arc A'YB' = ($\frac{\theta}{360} * 2\pi r$)

AXB = ($\frac{75}{360} * 2\pi r$)

A'YB' = ($\frac{35}{360} * 2\pi r$)

Ratio is,

AXB : A'YB'

Which is,

AXB = A'YB'

($\frac{75}{360} * 2\pi r$) = ($\frac{35}{360} * 2\pi r$)

360° on both sides get cancelled.

$2\pi r$ on both sides get cancelled.

Thus,

75°=25°

3=1

3:1

The ratio of arcs AXB: A'YB' is 3:1

Answer 2

Given:

Two circles have the center O and O'.

To Find:  

The ratio of arcs AXB and AYB

Solution:

The radius of the two circles is equal because circles are congruent.

Length of arc AXB = θ/360°× 2πr

                               = 70/360° × 2πr

Length of arc A'YB' = θ/360° × 2πr

                                = 25/360° × 2πr

Required Ratio = (75°/360° × 2πr) : (25°/360° × 2πr)

                         = 3 : 1

Therefore, the ratio of arcs AXB and AYB = 3 : 1