Question

Ares of two congruent circles subtend angles of 60 and 20 degrees at their respective centres,The ratio of lengths of corresponding ares is​

Answer 1

Answer:

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

Hope it will helps you

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⟹r

1

=

4

r

2

5

⟹

r

2

r

1

=

4

5

Answer 2

Given: Arcs of two congruent circles subtend angles of 60 and 20 degrees at their respective centres.

To find: Ratio of lengths of corresponding arcs

Solution: Two congruent circles have equal radii. Therefore, here radius of both the circles are equal.

Let the length of circle which subtends 60° be l1 and length of circle which subtends 20° be l2.

Now, we need to convert the angles given in degrees to radians. To convert degree into radians, we multiply it with π/180.

Therefore,

a1 =60°

= 60 × π/180 rad

= π/3 rad

a2=20°

= 20 × π/180 rad

= π/9 rad

The formula when arc subtends angle a at centre is:

length= radius × angle

Since r is equal, length is directly proportional to angle.

Therefore,

$\frac{l1}{l2} = \frac{a1}{a2}$

$= > \frac{l1}{l2} = \frac{ \frac{\pi}{3} }{ \frac{\pi}{9} }$

$= > \frac{l1}{l2} = \frac{9}{3}$

$= > \frac{l1}{l2} = 3$

=> l1:l2 = 3:1

Therefore, the ratio of lengths of corresponding arcs is 3:1 respectively.