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)|)))
wherex 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"
