Two circles are concentric. Two arcs are formed by the same central angle, with arc lengths of 54 an

Potherat8mi 2022-03-03 Answered
Two circles are concentric. Two arcs are formed by the same central angle, with arc lengths of 54 and 81 respectively. Which of the following are possible radii for the circles? so there are two possible answers.
- 14 and 41
- 23 and 32
- 32 and 64
- 40 and 60
- 12 and 18
I believe 12 and 18 are one.
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Expert Answer

Vikki Chapman
Answered 2022-03-04 Author has 8 answers
Consider r1= radius of inner circle
r2= radius of outer circle
S1= arc length of inner circle
S2= arc length of outer circle.
θ= central angle for both arcs S1 and S2.
relation s,r and θ is
S=rθ
S1=r1θ and S2=r2θr1=S1θ and r2=S2θ
Let r1r2=S1θS2θ=S1S2=5481=27×227×3=23
Since S1=54
S2=81
We have radius ratio 23
We check whether which options satisfy ratio
1) r1=14,r2=41r1r2=144123
2) r1=23,r2=32r1r2=233223
3) r1=40,r2=64r1r2=3264=16×216×4=2×12×2=1223
4) r1=40,r2=60r1r2=4060=46=2×22×3=23
5) r1=12,r2=18r1r2=1218=6×26×3=
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