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 18I believe 12 and 18 are one.

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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 18I believe 12 and 18 are one.

Vikki Chapman · Accepted Answer

Consider r1= radius of inner circler2= radius of outer circleS1= arc length of inner circleS2= arc length of outer circle.θ= central angle for both arcs S1 and S2.relation s,r and θ isS=rθ∴S1=r1θ and S2=r2θ⇒r1=S1θ and r2=S2θLet r1r2=S1θS2θ=S1S2=5481=27×227×3=23Since S1=54S2=81We have radius ratio 23We check whether which options satisfy ratio1) r1=14,r2=41⇒r1r2=1441≠232) r1=23,r2=32⇒r1r2=2332≠233) r1=40,r2=64⇒r1r2=3264=16×216×4=2×12×2=12≠234) r1=40,r2=60⇒r1r2=4060=46=2×22×3=235) r1=12,r2=18⇒r1r2=1218=6×26×3=

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

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