A circle's center is at #(2 ,1 )# and it passes through #(3 ,2 )#. What is the length of an arc covering #(7pi ) /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 #(7pi ) /12  # radians on the circle?

James Atkins · Accepted Answer

arc length#=(11sqrt(2))/6# units

Let A#(2,1)# be the center of the circle and #B(3,2)# be the arc length. Now, radius of circle#=AB=sqrt((3-2)^2+(2-1)^2)=sqrt(2)# #rarrtheta=#arc length/(radius #rarr=(7pi)/12=l/sqrt2# #rarr7*(22/7)*1/12=l/sqrt(2)# #rarrl=(11sqrt2)/6# units

Savannah Aguilar · Answer

undefined To find the length of an arc covering $ \frac{7\pi}{12} $ radians on the circle, we first need to find the radius of the circle using the given center and a point on the circle.

The distance formula can be used to find the radius of the circle, which is the distance between the center $(2, 1)$ and the point $(3, 2)$.

Then, once the radius is known, the formula for the length of an arc of a circle is used:

$$ \text{Arc Length} = r \times \theta $$

Where:
$ r $ = radius of the circle
$ \theta $ = angle in radians

Substitute the values into the formula to find the length of the arc.

Let's go step by step:

1. Find the radius:
$$ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

2. Calculate the angle in radians $ \theta $:
$$ \frac{7\pi}{12} $$

3. Use the formula for the length of an arc:
$$ \text{Arc Length} = r \times \theta $$

Plug in the values for $ r $ and $ \theta $ into the formula to get the length of the arc.