Question

If the length of an arc of a circle of radius r is equal to that of an arc of a
circle of radius 2r, then the angle of the corresponding sector of the first
circle is double the angle of the corresponding sector of the other circle.
Is the statement true or false? Justify your answer.​

Answer 1

Answer:

Consider two circles C

1

and C

2

of radii r and 2r respectively.

Let the length of two arcs be l

1

and l

2

.

l

1

=

AB

of C

1

=

360

∘

2πrθ

1

l

2

=

CD

of C

2

=

360

∘

2πrθ

2

=

360

∘

2π.2rθ

2

According

l

1

=l

2

⇒

360

∘

2πrθ

1

=

360

∘

2π.2rθ

2

⇒θ

1

=2θ

2

i.e., Angle of sector of the 1st circle is twice the angle of the sector of the other circle.

Hence, the given statement is true.

Answer 2

Consider two circles C

1

and C

2

of radii r and 2r respectively.

Let the length of two arcs be l

1

and l

2

.

l

1

=

AB

of C

1

=

360

∘

2πrθ

1

l

2

=

CD

of C

2

=

360

∘

2πrθ

2

=

360

∘

2π.2rθ

2

According

l

1

=l

2

⇒

360

∘

2πrθ

1

=

360

∘

2π.2rθ

2

⇒θ

1

=2θ

2

i.e., Angle of sector of the 1st circle is twice the angle of the sector of the other circle.

Hence, the given statement is true.

HOPE IT HELPS....☺️