Question

If in two circles, arcs of the same length subtend angles 45° and 75° at the centre of the circles and the radius of the first circle is 12 cm, then find the radius of the second circle.​

Answer 1

Step-by-step explanation:

Let the radii of the two circles be r

1

and r

2

. Let an arc of length I subtend an angle of 60

∘

at the centre of the circle of radius r

1

, while let an arc of length I subtend an angle of 75

∘

at the centre of the circle of radius r

2

.

Now, 60

∘

=

3

π

radian and 75

∘

=

12

5π

radian

We know that in a circle of radius r unit, if an arc of length l unit subtends an angle θ radian at the centre, then

θ=

r

l

or l=rθ

∴l=

3

r

1

π

and l=

12

r

2

5π

⟹

3

r

1

π

=

12

r

2

5π

⟹r

1

=

4

r

2

5

⟹

r

2

r

1

=

4

5