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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