The areas of two sectors of two different circles with equal corresponding arc lengths are equal. Is this statement true? If no, give reasons in support of your answer
Answers
Answer:
Given:
The radius of the circle C
1
=r
1
and
The radius of the circle C
2
=r
2
.
The length l
1
of an arc of C
1
= The length l
2
of an arc of C
2
.
The angle of the corresponding sector of C
1
=θ
1
and
The angle of the corresponding sector of C
2
=θ
2
To find out:
The validity of the statement that the area of the corresponding sector of C
1
= The area of the corresponding sector of C
2
.
Solution:
We know that the length of the arc of a sector of angle θ=
360
o
θ
×2πr, where r is the radius of the circle.
∴l
1
=
360
o
θ
1
×2πr&l
2
=
360
o
θ
2
×2π×2r.
Since l
1
=l
2
, we have
360
o
θ
1
×2πr=
360
o
θ
2
×2π×2r⇒θ
1
=2θ
2
....(i)
∴ Area of corresponding sector of C
1
=
360
o
θ
1
×πr
1
2
=
360
o
2θ
2
×πr
1
2
(from i) ....(ii)
and
Area of corresponding sector of C
2
=
360
o
θ
2
×πr
2
2
...(iii).
Comparing (ii) & (iii), we have
Area of corresponding sector of C
2
Area of corresponding sector ofC
1
=
360
o
θ
2
×πr
2
2
360
o
2θ
2
×πr
1
2
=
r
2
2
2r
1
2
.
So, the areas will be equal if and only if 2r
1
2
=r
2
2
.
So, the given statement is not correct.
