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Note: Figure not drawn to scale.
The circle shown has center O, circumference 144, and diameters PR
and QS. The length of arc PS is twice the length of arc PQ. What is the
length of arc QR?
Α. 24π
Β. 48π
C. 72π
D. 96π
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Let's solve this circle problem. We have a circle with center O, circumference 144π, and diameters PR and QS. We're told that the length of arc PS is twice the length of arc PQ, and we need to find the length of arc QR. Let's start by assigning variables to the central angles. If we call the central angle for arc PQ as θ, then the central angle for arc PS is 2θ. Since PR and QS are diameters, the angles are vertically opposite, so the central angle for arc RS is also θ, and the central angle for arc QR is 2θ.
To solve this problem, we need to use what we know about the relationship between arc length and central angle. The arc length equals the central angle in radians times the radius. Since the circumference is 144π, the radius is 72. We know that arc PS is twice arc PQ, so 3θ = 2θ + θ, which gives us θ = π/3. Therefore, the central angle for arc QR is 2θ = 2π/3. The arc length QR is then r times 2π/3, which is 72 times 2π/3, which equals 48π. So the answer is B: 48π.
Now, let's find the value of theta. We know that the sum of all central angles around a point is 360 degrees. Looking at our diagram, we have theta for arc PQ, 2 theta for arc QR, theta for arc RS, and 2 theta for arc SP. Adding these up, we get theta plus 2 theta plus theta plus 2 theta equals 360 degrees. This simplifies to 6 theta equals 360 degrees. Solving for theta, we get theta equals 60 degrees, which is pi over 3 radians. Therefore, the central angle for arc QR is 2 theta, which equals 120 degrees or 2 pi over 3 radians.
Now that we know the central angle for arc QR is 2 pi over 3 radians, we can calculate its length. The arc length is given by the formula: arc length equals radius times the central angle in radians. We'll focus on this in the next step.
Now, let's calculate the radius of the circle. We know that the circumference is 144π. The formula for circumference is 2π times the radius. So, 2π times r equals 144π. Solving for r, we get r equals 72. Now we can calculate the length of arc QR. The formula for arc length is radius times the central angle in radians. We found that the central angle for arc QR is 2π over 3 radians. So, the arc length QR equals 72 times 2π over 3, which simplifies to 144π over 3, or 48π. Therefore, the answer is B: 48π.
Let's verify our answer. We know that the central angle for arc QR is 120 degrees, which is one-third of the full 360 degrees. So, arc QR is one-third of the circumference. The circumference is 144π, so one-third of that is 48π. This confirms our answer: B. 48π.
Let's solve this problem using a different approach. We're given that arc PS is twice the length of arc PQ. Let's denote the length of arc PQ as x. Then, arc PS equals 2x. Since PR and QS are diameters, the angles are vertically opposite. This means arc RS equals arc PQ, which is x. Similarly, arc QR equals arc SP, which is 2x. The total circumference is the sum of all arcs: x plus 2x plus x plus 2x, which equals 6x. We know the circumference is 144π, so 6x equals 144π. Solving for x, we get x equals 24π. Therefore, arc QR equals 2x, which is 48π. So the answer is B: 48π.
This alternative approach gives us the same answer as our previous method. By using the relationship between the arcs and the total circumference, we found that the length of arc QR is 48π. This confirms our answer: B. 48π.
Let's summarize the key insights from this problem. First, when we have diameters in a circle, they create vertically opposite central angles. This helped us determine that if arc PQ subtends an angle theta, then arc RS also subtends the same angle theta. Similarly, if arc PS subtends an angle 2 theta, then arc QR also subtends an angle 2 theta. Second, we know that the sum of central angles around a point is 360 degrees. This allowed us to write the equation theta plus 2 theta plus theta plus 2 theta equals 360 degrees, which gave us theta equals 60 degrees. Third, we used the formula for arc length, which is radius times the central angle in radians. Alternatively, we can express arc length as the central angle divided by 360 degrees, multiplied by the circumference. Using either approach, we found that the length of arc QR is 48π. Therefore, the answer is B: 48π.
This problem demonstrates the power of using geometric properties and relationships to solve circle problems. By understanding the relationship between central angles and arc lengths, we were able to determine that the length of arc QR is 48π. Remember that in a circle, the ratio of an arc length to the circumference is equal to the ratio of the corresponding central angle to 360 degrees. This is a fundamental principle that can be applied to many circle problems.