A circle's center is at #(4 ,2 )# and it passes through #(6 ,7 )#. What is the length of an arc covering #(5pi ) /3 # radians on the circle?

Question

A circle's center is at #(4   ,2   )# and it passes through #(6    ,7   )#.  What is the length of an arc covering #(5pi ) /3  # radians on the circle?

Sophia Alfred · Accepted Answer

Arc length#~~28.2# to 1 decimal place

Let the radius of the circle be #r# Let the length of arc be #L_a# Distance from the circles centre to any point on its circumference is always the same. '~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(blue)("Determine the radius of the circle")# #=> r=sqrt((x_2-x_1)^2+(y_2-y_1)^2)# #=> r = sqrt((6-4)^2 +(7-2)^2)# #=>color(blue)(r = sqrt(29))" "-># 29 is a prime number so can not be simplified '~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(blue)("Determine the length of arc")# #color(brown)("Important point:")# #color(brown)("1 radian is such that its length of ark is the same as")# #color(brown)("the length of the radius.")# So the length of arc #L_a=rxx (5pi)/3# #=> L_a=sqrt(29)xx (5pi)/3# #color(blue)(L_a~~28.2" to 1 decimal place")# '~~~~~~~~~~~~~~~~~~~~~~~~~ #color(blue)("Check:")# Circumference = #piD = pixx2sqrt(29)# and we have #1 2/3" of "1/2 # of the circumference #=>L_a=1 2/3 xxpisqrt(29) =28.19...# Confirmed!

Sadie Allen · Answer

undefined To find the length of an arc covering $ \frac{5\pi}{3} $ radians on the circle, we first need to calculate the radius of the circle using the given center and a point on the circle. Then, we can use the formula for the length of an arc on a circle:

$$ \text{Arc Length} = r \times \text{angle in radians} $$

where $ r $ is the radius of the circle and the angle is given in radians.