A circle's center is at #(5 ,4 )# and it passes through #(1 ,4 )#. What is the length of an arc covering #(5pi ) /8 # radians on the circle?

Question

A circle's center is at #(5 ,4 )# and it passes through #(1    ,4    )#.  What is the length of an arc covering #(5pi ) /8  # radians on the circle?

Aurora Ayala · Accepted Answer

Use the distance formula, the the arc length formula.

First things first use the distance formula for two points: D= #sqrt((x_2-x_1)^2 + (y_2-y_1)^2# which boils down to: #sqrt((5-1)^2+(4-4)^2)=4# in this case. This will give you the radius of the circle. Since both points are at the same height, you didn't really need the distance formula, but hey, why not be general? Then use the arc length formula for a circle. Since the circumference is 2#pi#r, the length of an arc will just be the fraction of the circumference swept out by the angle given. Which is to say: #L=2pi*r# for the full circle, #L= r*t# for an arc, Where L is the arc length, r is the radius, and t is the central angle. This gives you an arc length of: L = #4*(5pi/8)# = #5pi/2#

Jayden Jackson · Answer

undefined To find the length of an arc on a circle, you can use the formula:

Arc Length = Radius × Angle in radians

Given that the circle's center is at (5, 4) and it passes through (1, 4), the radius of the circle is the distance between the center and any point on the circle, which can be calculated using the distance formula.

Using the distance formula:

Radius = √((5 - 1)^2 + (4 - 4)^2)
       = √(4^2 + 0^2)
       = √16
       = 4

Now, we can use the formula for arc length:

Arc Length = Radius × Angle in radians
            = 4 × (5π/8)
            = 5π units