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

3 Answers
Feb 11, 2018

the shortest arc happens to be #s=4.957#

Explanation:

Given:
#(5,4)# is a point on the circle
#(2,2)#is a point on the circle

Angle subtended at the center is #(5pi)/4# which is reflex lying on the major arc
The minor arc happens to be #2pi-(5pi)/4=(3pi)/4#

The triangle formed by the two points and the center happens to be an isoceles triangle with the shortest angle formed at the center being #alpha=(3pi)/4#.

Remaining angle is
#(pi-(3pi)/4)=pi/4#

Angle at each of the vertex at the base is #theta=1/2(pi)/4=pi/8#
Mid point happens to be
#((5+2)/2,(4+2)/2)=(3.5,3)#

We calculate the radius of circle by pythagoras theorem in the right angled triangle formed by one of the point, say #(5,4)#, the mid-point #(3.5,3)#, and the center of the circle.

Distance between one of the points, say #(5,4)# and the mid point #(3.5,3)# is given by
#a^2=(3.5-5)^2+(3-4)^2=(-1.5)^2+(-1)^2#
#= 2.25+1=3.25#
#a^2=3.25#

Also, the line joining #(5,4) and (3.5,3)# happens to be the adjacent side and the radius being hypotenuse for the angle at the point #(5,4)#

Consideing the ratio #costheta#,
#cos^2theta=a^2/r^2#
#theta=pi/8#
#a^2=3.25#

#cos^2(pi/8)=3.25/r^2#
#cos^2(pi)/8=0.146#
#0.854=3.25/r^2#
#r^2=3.25/0.854=3.808#
#r=sqrt3.808#
#r=1.951#

Hence, the shortest arc happens to be
#s=ralpha=1.951(3pi)/4#
#s=1.533#
the shortest arc happens to be #s=4.957#

Feb 11, 2018

Shortest Arc length #s = 4.5946#

Explanation:

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#vec(AD) = sqrt(5-2)^2 + (4-2)^2) = 3.6056#

#r = (AD) / (2 sin (theta/2)) = 3.6056 / (2 sin ((5pi)/8)) = 1.95#

Since the center angle #theta# is more than #pi#, to get the shortest arc length we must subtract from #(2pi)#

Shortest length of the arc

#s = r * theta = 1.95 * (2pi - ((5pi)/4)) = 1.95 * ((3pi)/4) = 4.5946#

Feb 11, 2018

Shorter arc length between the points is #4.59 # unit.

Explanation:

Angle subtended at the center by the arc is #theta_1=(5pi)/4#

Angle subtended at the center by the minor arc is

# theta=2pi-(5pi)/4= (3pi)/4 :. theta/2 =(3pi)/8 #

Distance between two points #(5,4) and (2,2) # is

#D= sqrt ((x_1-x_2)^2+(y_1-y_2)^2) =sqrt ((5-2)^2+(4-2)^2# or

#D=sqrt13 ~~ 3.61# unit. Chord length #l ~~3.61# unit.

Formula for the length of a chord is #L_c= 2r sin (theta/2)#

where #r# is the radius of the circle and #theta# is the angle

subtended at the center by the chord.

#:. 3.61=2*r*sin((3pi)/8) or r = 3.61/(2*sin((3pi)/8)# or

#r= 3.61/1.85 ~~1.95:.# Radius of the circle is #1.95# unit.

Arc length is # L_a= cancel(2 pi )* r*theta/cancel(2pi)=r *theta# or

#L_a=1.95*(3pi)/4 ~~ 4.59 # unit .

Shorter arc length between the points is #4.59 # unit.[Ans]