A circle's center is at #(5 ,4 )# and it passes through #(1 ,4 )#. What is the length of an arc covering #(2pi ) /3 # radians on the circle?
1 Answer
≈ 8.378 units.
Explanation:
The length of the arc is found by calculating the fraction of the circumference.
That is
#color(red)(bar(ul(|color(white)(2/2)color(black)("arc" ="circumference" xxx/(2pi))color(white)(2/2)|)))#
where#x# is the angle subtended at the centre of the circle.To calculate circumference
#=2pir# we require to know r, the radius.We are given the coordinates of the centre and a point on the circumference. Hence the radius is the distance between these 2 points.
The 2 points (5 ,4) and (1 ,4) have the same y-coordinate and so they lie on a horizontal line ( y = 4).
The distance between the points is therefore the difference in the x-coordinates.
#rArr"radius" =r=5-1=4# length of arc
#=cancel(2pi)xx4xx((2pi)/3)/(cancel(2pi))#
#=(4xx2pi)/3≈8.378" units to 3 decimal places"#
