Question

Two arc of same length of two different circles subtend angles of 25 degree and 30 degree at their centres respectively the ratio of the radii of the circle is

Answer 1

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11th

Maths

Trigonometric Functions

Angle and its Measurement

If in two circles, arcs of ...

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Asked on October 15, 2019 by

Mansha Pooja

If in two circles, arcs of the same length subtend angles 60

∘

and 75

∘

at the centre, find the ratio of their radii.

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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

⟹r

1

=

4

r

2

5

⟹

r

2

r

1

=

4

5

Answer 2

Answer:

Ratio of radius of circles are $r_{1}: r_{2}=6: 5 .$

Explanation:

Let radii of circle r1 and r2.

Angle subtended by an arc at the centre of first circle is θ = 25° = 5π /36 radian

Angle subtended by an arc at the centre of second circle is = 30° = π /6 From formula :

Length of arc ( l ) = radius (r) x angle (θ)

∴ Length of arc of first circle = 5π /36 x r1

Length of arc of second circle = π /6 x r2

Given that: Arcs of two circles are of same length

Then$\quad \frac{5\pi}{36} \times r_{1}=\frac{ \pi}{6} \times r_{2}$

$\frac{r_1}{r_{2}}=\frac{36}{5\times 6}\\=\frac{6}{5}$

$\therefore \quad r_{1}: r_{2}=6: 5$

Hence, ratio of radius of circles are

$r_{1}: r_{2}=6: 5 .$