Given that circle Q has a radius of 42, and radii line QR and line QS form an angle of 180, find the exact length of arc RS.
To find the length of arc RS, we need to know the central angle that arc RS subtends. However, we only have information about the angle formed by radii line QR and line QS, which is a diameter.
Let's break down the problem into steps:
Step 1: Calculate the radius of circle Q
Given that the radius of circle Q is 42, we can use this information to find the circumference of the circle. The formula for the circumference of a circle is C = 2πr, where r is the radius.
C = 2π(42) = 84π
Step 2: Calculate the circumference of the entire circle
Since the angle formed by radii lines QR and QS is 180 degrees, it means that this angle is half of the full circle's angle (360 degrees). So if we calculate the circumference of arc RS, it will be half the circumference of the entire circle.
Circumference of the entire circle = 2πr = 84π
Step 3: Calculate the length of arc RS
The length of arc RS can be calculated using the formula for the length of an arc. The formula is L = (θ/360) * C, where L is the length of the arc, θ is the central angle, and C is the circumference of the circle.
Since the central angle formed by radii lines QR and QS is 180 degrees, we can substitute the values into the formula:
L = (180/360) * 84π
L = (1/2) * 84π
L = 42π
Therefore, the exact length of arc RS is 42π units.