Points #(3 ,2 )# and #(8 ,4 )# are #( pi)/3 # radians apart on a circle. What is the shortest arc length between the points?

Question

Points #(3   ,2    )# and #(8   ,4   )# are #(  pi)/3  # radians apart on a circle. What is the shortest arc length between the points?

Violet Aldrich · Accepted Answer

#s = (pisqrt29)/3#

The square of distance, d, between the two points is:

#d² = (8 - 3)² + (4 - 2)²#

#d² = (5)² + (2)²#

#d² = 29#

d forms one side of a triangle where the other two sides are radii; this allow us the law of cosines to find the radius:

#29 = r² + r² - 2(r)(r)cos(pi/3)##

#r² = 29/(2 - 2cos(pi/3))#

#r = sqrt29#

The arc length, s, is:

#s = rtheta#

#s = (pisqrt29)/3#

Theodore Abad · Answer

undefined The shortest arc length between the points (3, 2) and (8, 4) on a circle that are π/3 radians apart is 5π/3 units.

Noah Atkinson · Answer

undefined To find the shortest arc length between two points on a circle, we need to calculate the length of the arc along the circumference of the circle formed by connecting the two points.

First, we calculate the radius of the circle. We can use the distance formula to find the distance between the two points, which will be the diameter of the circle.

Using the points (3, 2) and (8, 4), the distance between them is $ \sqrt{(8 - 3)^2 + (4 - 2)^2} = \sqrt{5^2 + 2^2} = \sqrt{25 + 4} = \sqrt{29} $.

Since the distance between the points is the diameter of the circle, the radius is half of this distance, so the radius is $ \frac{\sqrt{29}}{2} $.

Now, to find the arc length between the points that are $ \frac{\pi}{3} $ radians apart, we use the formula for the arc length of a circle, which is given by $ s = r \cdot \theta $, where $ r $ is the radius and $ \theta $ is the central angle in radians.

Substituting the values, we get $ s = \frac{\sqrt{29}}{2} \cdot \frac{\pi}{3} $.

Therefore, the shortest arc length between the points is $ \frac{\sqrt{29}}{2} \cdot \frac{\pi}{3} $.