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

Related questions
See all questions in Circle Arcs and Sectors
Impact of this question
1402 views around the world
You can reuse this answer
Creative Commons License