A circle's center is at #(7 ,4 )# and it passes through #(8 ,2 )#. What is the length of an arc covering #( pi ) /6 # radians on the circle?

Question

A circle's center is at #(7   ,4   )# and it passes through #(8    ,2   )#.  What is the length of an arc covering #( pi ) /6  # radians on the circle?

Theodore Abad · Accepted Answer

#root2(5)pi/6# The radius is #AP=root2((7-8)^2+(4-2)^2)=root2(5)# As the by definition an angle in radiants is exactly the ratio between the arc and the radius of the circumference where that arc is determined by that angle itself, we #pi/6=#("circular arc")#/R#. As a result the arc length is "circular arc"#=R*pi/6#

Allison Allen · Answer

undefined To find the length of an arc covering $ \frac{\pi}{6} $ radians on the circle, you first need to find the radius of the circle using the given points. Then, you can use the formula for the arc length of a circle, which is $ s = r \theta $, where $ s $ is the arc length, $ r $ is the radius of the circle, and $ \theta $ is the angle in radians subtended by the arc.

1. Calculate the radius $ r $ using the distance formula between the center of the circle $ (x_1, y_1) $ and a point on the circle $ (x_2, y_2) $.
$$ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

2. Substitute the values into the formula for arc length: 
$$ s = r \theta $$

3. Calculate the arc length $ s $ using $ \frac{\pi}{6} $ radians as $ \theta $.

4. Round the result to an appropriate number of decimal places if needed.