A circle's center is at #(2 ,1 )# and it passes through #(3 ,2 )#. What is the length of an arc covering #(5pi ) /12 # radians on the circle?

Question

A circle's center is at #(2 ,1 )# and it passes through #(3    ,2   )#.  What is the length of an arc covering #(5pi ) /12  # radians on the circle?

James Atkins · Accepted Answer

Length of arc covering #(5pi)/12# radian on circle is #1.851# As the circle's center is at #(2,1)# and it passes through #(3,2)#, the distance between these two points will give the radius. Hence, radius is #sqrt((3-2)^2+(2-1)^2)=sqrt(1^2+1^2)=sqrt2=1.4142# Hence, length of arc covering #(5pi)/12# radian on circle is #(5pi)/12xx1.4142=(5xx3.1416xx1.4142)/12=1.851#

Sadie Allen · Answer

undefined The length of an arc on a circle is given by the formula $ s = r \cdot \theta $, where $ s $ is the arc length, $ r $ is the radius of the circle, and $ \theta $ is the central angle in radians.

Given that the circle's center is at (2, 1) and it passes through (3, 2), we can find the radius using the distance formula:

$$
r = \sqrt{(3 - 2)^2 + (2 - 1)^2} = \sqrt{1 + 1} = \sqrt{2}
$$

The angle $ \theta $ corresponding to $ \frac{5\pi}{12} $ radians can be calculated as:

$$
\theta = \frac{5\pi}{12}
$$

Thus, the length of the arc is:

$$
s = (\sqrt{2}) \cdot \frac{5\pi}{12} \approx 1.307 \text{ units}
$$