Points #(4 ,4 )# and #(7 ,3 )# are #(5 pi)/4 # radians apart on a circle. What is the shortest arc length between the points?

Question

Points #(4  ,4   )# and #(7   ,3    )# are #(5  pi)/4    # radians apart on a circle. What is the shortest arc length between the points?

Sophia Alfred · Accepted Answer

#S = (3pi)/4sqrt(5/3(2-sqrt2))#

Because the angle #(5pi)/4# is greater than #pi#, we know that this is NOT the smallest angle between the two points; the smallest angle is:

#angle theta=2pi-(5pi)/4#

#angle theta = (3pi)/4#

This will be the angle between the two radii that connect the two points #(4,4)# and #(7,3)#. The two radii and the chord, c, between the two points form a triangle, therefore, we can write an equation, using the Law of Cosines:

#c^2 = r^2 + r^2-2(r)(r)cos(theta)#

We know that #c^2# is the square of the distance between the two points and we know the value of #theta#:

#(7-4)^2+(3-4)^2 = r^2 + r^2-2(r)(r)cos((3pi)/4)#

Remove a common factor of #r^2#:

#3^2+(-1)^2= r^2(2-2cos((3pi)/4))#

Evaluate the cosine function:

#10= r^2(2-2(-sqrt2/2))#

Simplify:

#10= r^2(2+sqrt2)#

Multiply both sides by the conjugate:

#10(2-sqrt2)= r^2(2+sqrt2)(2-sqrt2)#

#10(2-sqrt2)= 6r^2#

Flip the equation and divide both sides by 6:

#r^2 = 5/3(2-sqrt2)#

Use the square root operation on both sides:

#r = sqrt(5/3(2-sqrt2))#

We know that the arclength, S, is the radius multiplied by the radian measure of angle:

#S = thetar#

#S = (3pi)/4sqrt(5/3(2-sqrt2))#