Points (5 ,2 ) and (1 ,4 ) are (5 pi)/3 radians apart on a circle. What is the shortest arc length between the points?

1 Answer
Yonas Yohannes
Mar 14, 2016

Arc Length, L = r theta = 2sqrt(5/3)*pi/3 = (2pi)/3*sqrt(5/3)

Explanation:

Take a look at the points in the circle in the figure:
The chord bar(AB) can be calculated the distance formula:
AB = sqrt((5-1)^2 + (2-4)^2) = sqrt(16+4)=sqrt(20)=2sqrt(5)
Now we know the outside angle between bar(OB) and bar(OA)
angle alpha=BhatOA = (5pi)/3 thus the inside angle theta=BhatOA = (pi)/3
From the picture we can see that bar(BO) = bar(AO) = r = Radius
Also bar(AB) = bar(AH) + bar(HB) = sqrt(5)

Now triangle AOH form a right angle triangle with angles:
/_HAO = beta = pi/3

/_AOH = theta/2 = pi/6

This the 30-60-90 right angle triangle with sides relationship of
(2:1 :sqrt(3))
Using (AOH)_(Delta) we write bar(AO) = r = bar(AH)/sin(beta) = sqrt(5)/sinbeta
Now =>sinbeta = sqrt(3)/2
Thus r = 2 sqrt(5)/sqrt(3)= 2sqrt(5/3)
Now Arc Length, L = r theta = 2sqrt(5/3)*pi/3 = (2pi)/3*sqrt(5/3)
enter image source here

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