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]