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? Select the two correct answers.
40 and 60
23 and 32
32 and 64
14 and 41
12 and 18
To find the possible radii for the circles, we need to use the given information that two arcs are formed by the same central angle, with arc lengths of 54 and 81 respectively.
Let's denote the central angle as θ and the radius of the circles as r.
The formula to calculate the length of an arc is given by:
Arc Length = (θ / 360) * (2 * π * r)
We can rearrange the formula to solve for the angle θ:
θ = (Arc Length * 360) / (2 * π * r)
Now, for the given arc lengths of 54 and 81:
For the first arc length of 54:
θ1 = (54 * 360) / (2 * π * r) = (19440 / πr) degrees
For the second arc length of 81:
θ2 = (81 * 360) / (2 * π * r) = (29160 / πr) degrees
Since the two arcs are formed by the same central angle, we equate θ1 and θ2:
(19440 / πr) = (29160 / πr)
We can simplify this equation by canceling out πr from both sides:
19440 = 29160
This equation is not possible since 19440 is not equal to 29160. Therefore, there are no valid radii that satisfy the given conditions.
Hence, none of the options provided (40 and 60, 23 and 32, 32 and 64, 14 and 41, 12 and 18) are correct.
Angle = arc/radius
81/R1 = 54/R2
54 R1 = 81 R2
R1/R2 = 81/54 = 9*9 / 9*6 = 3*3*3*3 / 3*3*3*2 = 3/2
60/40 = 20*3 / 20*2 = 3/2 sure enough
now it's your turn( If you can not divide the big one by 3, it is no good.)