A circle has a center at #(7 ,6 )# and passes through #(2 ,1 )#. What is the length of an arc covering #pi/8# radians on the circle?

1 Answer
Sudip Sinha
Jan 27, 2016

# ( 5 sqrt(2) pi ) / 8 #

Explanation:

We know the center and a point on the circle. The distance between the two points is the radius (draw a picture and convince yourself this).

We know that the distance between two points # (x_1, y_1) # and # (x_2, y_2) # on the Euclidean plane is given by # d = sqrt( (x_1 - x_2)^2 + (y_1 - y_2)^2 ) #.
Thus, radius # r = sqrt( (7 - 2)^2 + (6 - 1)^2 ) = sqrt(5^2 + 5^2) = sqrt(2 * 5^2) = 5 sqrt(2) #.

By definition, a radian is the angle subtended by an arc of length equal to the radius. Thus, an arc which subtends an angle of # theta # has a length of # theta r #. In this case, # theta = pi / 8 #, so arc length = # ( 5 sqrt(2) pi ) / 8 #.

Hint to remember
Note that the circumference subtends and angle of # 2 pi # at the center, and has a length of # 2 pi r #. By unitary method, an arc which subtends an angle of # theta # has a length of # theta r #.

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