Points (5, 4) and (2, 2) are 5π4 radians apart on a circle. What is the shortest arc length between the points?

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Vaughn Bradley · Accepted Answer

Step 1  Given:  (5, 4) is a point on the circle  (2, 2) is a point on the circle  Angle subtended at the center is 5π4 which is reflex lying on the major arc  The minor arc happens to be  2π−5π4=3π4  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  α=3π4  Remaining angle is  (π−3π4)=π4  Angle at each of the vertex at the base is  θ=12π4=π8  Mid point happens to be  (5+22, 4+22)=(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  a2=(3.5−5)2+(3−4)2=(−1.5)2+(−1)2  =2.25+1=3.25  a2=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 cos⁡θ  cos2θ=a2r2  θ=π8  a2=3.25  cos2(π8)=3.25r2  cos2(π)8=0.146  0.854=3.25r2  r2=3.250.854=3.808  r=3.808  r=1.951  Hence, the shortest arc happens to be  s=rα=1.9513π4  s=1.533  the shortest arc happens to be s=4.957

limegreenleosp9 · Accepted Answer

Step 1  Angle subtended at the center by the arc is  θ1=5π4  Angle subtended at the center by the minor arc is  θ=2π−5π4=3π4∴θ2=3π8  Distance between two points (5, 4) and (2, 2) is  D=(x1−x2)2+(y1−y2)2=(5−2)2+(4−2)2  or  D=13≈3.61  unit. Chord length  l≈3.61 unot  Formula for the length of a chord is  Lc=2rsin⁡(ta^2)  where r is the radius of the circle and θ is the angle subtended at the center by the chord.  ∴3.61=2×r×sin⁡(3π8)  or  r=3.612×sin⁡(3π8)  or  r=3.611.85≈1.95∴ Radius of the circle is 1.95 unit  Arc length is  Undefined control sequence cancelUndefined control sequence cancel  or  La=1.95×3π4≈4.59 unit  Shorter arc length between the points is 4.59 unit.

Points (5, 4) and (2, 2) are \frac{5\pi}{4} radians apart

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