Points $(3, 2)$ $(3 ,2 )$ and $(8, 1)$ $(8 ,1 )$ are $\frac{2 π}{3}$ $(2 pi)/3$ radians apart on a circle. What is the shortest arc length between the points?

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Points $(3, 2)$ $(3 ,2 )$ and $(8, 1)$ $(8 ,1 )$ are $\frac{2 π}{3}$ $(2 pi)/3$ radians apart on a circle. What is the shortest arc length between the points?

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Answer 1 · 2016-03-10T03:13:05

Shortest arc $s = 6.16576$ $s=6.16576" "$ units

Explanation:

From the given data: Points (3,2) and (8,1) are 2π/3 radians apart on a circle. What is the shortest arc length between the points?

This means , we have the central angle $θ = \frac{2 π}{3}$ $theta=(2pi)/3$ . To determine the arc $s$ $s$ , we need to know the radius $r$ $r$ .

Let us solve $r$ $r$ :
Let $(x, y)$ $(x, y)$ be the center of the circle

$r = r$ $r=r$

${(3 - x)}^{2} + {(2 - y)}^{2} = {(8 - x)}^{2} + {(1 - y)}^{2}$ $(3-x)^2+(2-y)^2=(8-x)^2+(1-y)^2$
After simplifying this equation, we have

$5 x - y - 26 = 0$ $color(red)(5x-y-26=0" ")$ $first equation$ $color(blue)("first equation")$

Making use of the slopes of the radii:

Let $m_{2} = \frac{y - 1}{x - 8}$ $m_2=(y-1)/(x-8)$ and $m_{1} = \frac{y - 2}{x - 3}$ $m_1=(y-2)/(x-3)$

$tan θ = tan (\frac{2 π}{3}) = \frac{m_{2} - m_{1}}{1 + m_{2} \cdot m_{1}}$ $Tan theta=Tan ((2pi)/3)=(m_2-m_1)/(1+m_2*m_1)$

$tan (\frac{2 π}{3}) = - \sqrt{3} = \frac{\frac{y - 1}{x - 8} - \frac{y - 2}{x - 3}}{1 + \frac{y - 1}{x - 8} \cdot \frac{y - 2}{x - 3}}$ $Tan((2pi)/3)=-sqrt3=((y-1)/(x-8)-(y-2)/(x-3))/(1+(y-1)/(x-8)*(y-2)/(x-3))$

After simplifying this equation

$x + 5 y - 13 = - \sqrt{3} \cdot (x^{2} + y^{2} - 11 x - 3 y + 26)$ $color(red)(x+5y-13=-sqrt3*(x^2+y^2-11x-3y+26))$ $second equation$ $color(blue)("second equation")$

Use now, the first and second equations to solve for the center (x, y)

We have

$26 \sqrt{3} x^{2} + (26 - 286 \sqrt{3}) x + 780 \sqrt{3} - 143 = 0$ $color(red)(26sqrt3x^2+(26-286sqrt3)x+780sqrt3-143=0)$

By Quadratic Equation Formula

$x = 5.78868$ $x=5.78868$ and $y = 2.9434$ $y=2.9434$

Compute now for radius $r$ $r$ using center (x, y) and point on the circle (8, 1)

$r = \sqrt{{(8 - 5.78868)}^{2} + {(1 - 2.9434)}^{2}}$ $r=sqrt((8-5.78868)^2+(1-2.9434)^2)$

$r = 2.94393$ $r=2.94393" "$ units

Compute now for the arc

$s = r \cdot θ = 2.94393 \cdot \frac{2 π}{3} = 6.16576$ $s=r*theta=2.94393*(2pi)/3=6.16576" "$ units

God bless....I hope the explanation is useful.

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Points $(3, 2)$ $(3 ,2 )$ and $(8, 1)$ $(8 ,1 )$ are $\frac{2 π}{3}$ $(2 pi)/3$ radians apart on a circle. What is the shortest arc length between the points?

1 Answer

Explanation:

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Points (3,2)(3 ,2 ) and (8,1)(8 ,1 ) are 2π3(2 pi)/3 radians apart on a circle. What is the shortest arc length between the points?

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Points $(3, 2)$ $(3 ,2 )$ and $(8, 1)$ $(8 ,1 )$ are $\frac{2 π}{3}$ $(2 pi)/3$ radians apart on a circle. What is the shortest arc length between the points?