1.Two circles are concentric, and have radii of 15 and 25 respectively. If two arcs are formed by the same central angle, which of the following are possible arc lengths? Select the two correct answers.
-30 and 40
-3 and 5
-60 and 90
-45 and 75
-50 and 80
2.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.
-12 and 18
-23 and 32
-32 and 64
-40 and 60
-14 and 41
#1. the ratio is 15:25 = 3:5
#2. The ratio is 54:81 = 6:9
1. To find the possible arc lengths, we need to use the formula for arc length:
Arc Length = Central Angle * Radius
Since the central angle is the same for both arcs, we can write the equation as:
Arc Length1 = Arc Length2
Substituting the formula for arc length, we get:
Central Angle * Radius1 = Central Angle * Radius2
Since the central angle is the same, we can cancel it out:
Radius1 = Radius2
So, the arc lengths of the two circles will be the same if the radii are the same. Looking at the options, the only possible radii that are the same are 15 and 25. Therefore, the possible arc lengths are:
-30 and 40
2. To find the possible radii for the circles, we need to use the formula for arc length:
Arc Length = Central Angle * Radius
Given that two arcs are formed by the same central angle, but with arc lengths of 54 and 81, we can set up the following equation:
Central Angle * Radius1 = 54
Central Angle * Radius2 = 81
Since the central angle is the same in both equations, we can cancel it out:
Radius1 = 54
Radius2 = 81
Looking at the options, we can see that the possible radii are 32 and 64. Therefore, the correct answers are:
-32 and 64