Points $(5, 4)$ $(5 ,4 )$ and $(2, 2)$ $(2 ,2 )$ are $\frac{5 π}{4}$ $(5 pi)/4$ radians apart on a circle. What is the shortest arc length between the points?

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Points $(5, 4)$ $(5 ,4 )$ and $(2, 2)$ $(2 ,2 )$ are $\frac{5 π}{4}$ $(5 pi)/4$ radians apart on a circle. What is the shortest arc length between the points?

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Answer 1 · 2018-02-11T11:06:40

the shortest arc happens to be $s = 4.957$ $s=4.957$

Explanation:

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

Angle subtended at the center is $\frac{5 π}{4}$ $(5pi)/4$ which is reflex lying on the major arc
The minor arc happens to be $2 π - \frac{5 π}{4} = \frac{3 π}{4}$ $2pi-(5pi)/4=(3pi)/4$

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 $α = \frac{3 π}{4}$ $alpha=(3pi)/4$ .

Remaining angle is
$(π - \frac{3 π}{4}) = \frac{π}{4}$ $(pi-(3pi)/4)=pi/4$

Angle at each of the vertex at the base is $θ = \frac{1}{2} \frac{π}{4} = \frac{π}{8}$ $theta=1/2(pi)/4=pi/8$
Mid point happens to be
$(\frac{5 + 2}{2}, \frac{4 + 2}{2}) = (3.5, 3)$ $((5+2)/2,(4+2)/2)=(3.5,3)$

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

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

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

Consideing the ratio $cos θ$ $costheta$ ,
${cos}^{2} θ = \frac{a^{2}}{r^{2}}$ $cos^2theta=a^2/r^2$
$θ = \frac{π}{8}$ $theta=pi/8$
$a^{2} = 3.25$ $a^2=3.25$

${cos}^{2} (\frac{π}{8}) = \frac{3.25}{r^{2}}$ $cos^2(pi/8)=3.25/r^2$
$\frac{{cos}^{2} (π)}{8} = 0.146$ $cos^2(pi)/8=0.146$
$0.854 = \frac{3.25}{r^{2}}$ $0.854=3.25/r^2$
$r^{2} = \frac{3.25}{0.854} = 3.808$ $r^2=3.25/0.854=3.808$
$r = \sqrt{3.808}$ $r=sqrt3.808$
$r = 1.951$ $r=1.951$

Hence, the shortest arc happens to be
$s = r α = 1.951 \frac{3 π}{4}$ $s=ralpha=1.951(3pi)/4$
$s = 1.533$ $s=1.533$
the shortest arc happens to be $s = 4.957$ $s=4.957$

Answer 2 · 2018-02-11T11:51:06

Shortest Arc length $s = 4.5946$ $s = 4.5946$

Explanation:

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$- - \to A D = {\sqrt{5 - 2}}^{2} + {(4 - 2)}^{2}) = 3.6056$ $vec(AD) = sqrt(5-2)^2 + (4-2)^2) = 3.6056$

$r = \frac{A D}{2 sin (\frac{θ}{2})} = \frac{3.6056}{2 sin (\frac{5 π}{8})} = 1.95$ $r = (AD) / (2 sin (theta/2)) = 3.6056 / (2 sin ((5pi)/8)) = 1.95$

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

Shortest length of the arc

$s = r \cdot θ = 1.95 \cdot (2 π - (\frac{5 π}{4})) = 1.95 \cdot (\frac{3 π}{4}) = 4.5946$ $s = r * theta = 1.95 * (2pi - ((5pi)/4)) = 1.95 * ((3pi)/4) = 4.5946$

Answer 3 · 2018-02-11T12:16:50

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

Explanation:

Angle subtended at the center by the arc is $θ_{1} = \frac{5 π}{4}$ $theta_1=(5pi)/4$

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]

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Points $(5, 4)$ $(5 ,4 )$ and $(2, 2)$ $(2 ,2 )$ are $\frac{5 π}{4}$ $(5 pi)/4$ radians apart on a circle. What is the shortest arc length between the points?

3 Answers

Explanation:

Explanation:

Explanation:

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Points $(5, 4)$ $(5 ,4 )$ and $(2, 2)$ $(2 ,2 )$ are $\frac{5 π}{4}$ $(5 pi)/4$ radians apart on a circle. What is the shortest arc length between the points?