Points $(7, 1)$ $(7 ,1 )$ and $(8, 9)$ $(8 ,9 )$ are $\frac{3 π}{4}$ $(3 pi)/4$ radians apart on a circle. What is the shortest arc length between the points?

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Points $(7, 1)$ $(7 ,1 )$ and $(8, 9)$ $(8 ,9 )$ are $\frac{3 π}{4}$ $(3 pi)/4$ radians apart on a circle. What is the shortest arc length between the points?

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Answer 1 · 2016-07-19T12:26:36

$\frac{3 \sqrt{65}}{4 \sqrt{2 + \sqrt{2}}}$ $color(blue)((3sqrt(65))/(4sqrt(2+sqrt(2)))$

(This is pretty ugly; could someone check this please?)

Explanation:

Distance between $(7, 1)$ $(7,1)$ and $(8, 9)$ $(8,9)$ is given by the Pythagorean Theorem as
$XXX \sqrt{1^{2} + 8^{2}} = \sqrt{65}$ $color(white)("XXX")sqrt(1^2+8^2)=sqrt(65)$

To determine the arc length we will need to determine the radius of a circle with the given points at an angle of $\frac{3 π}{4}$ $(3pi)/4$ relative to the center of the circle.
enter image source here
(Note this diagram is not accurate).

If we let the radius of this circle be $r$ $color(green)(r)$
and denote
$XXX (7, 1)$ $color(white)("XXX")(7,1)$ as $P$ $P$ ,
$XXX (8, 9)$ $color(white)("XXX")(8,9)$ as $Q$ $Q$ ,
$XXX$ $color(white)("XXX")$ the center of the circle as $C$ $C$ , and
$XXX$ $color(white)("XXX")$ the point on the extension of $P C$ $PC$ to form a right angle with $Q$ $Q$ as $S$ $S$

then
Since
$XXX ∠ P S Q = \frac{π}{2}$ $color(white)("XXX")/_PSQ=pi/2$ , and
$XXX ∠ Q C S = π - \frac{3 π}{4} = \frac{π}{4}$ $color(white)("XXX")/_QCS=pi-(3pi)/4=pi/4$
$\to$ $rarr$
$XXX ∠ C Q S = \frac{π}{4}$ $color(white)("XXX")/_CQS=pi/4$ and
$XXX | C S | = | Q S | = \frac{r}{\sqrt{2}}$ $color(white)("XXX")abs(CS)=abs(QS)=r/sqrt(2)$

Therefore
$XXX | P S | = r + \frac{r}{\sqrt{2}} = (\frac{\sqrt{2} + 1}{\sqrt{2}}) r$ $color(white)("XXX")abs(PS)=r+r/sqrt(2)=((sqrt(2)+1)/sqrt(2))r$
and
$XXX | P Q | = \sqrt{{(\frac{r}{\sqrt{2}})}^{2} + {((\frac{\sqrt{2} + 1}{\sqrt{2}}) r)}^{2}}$ $color(white)("XXX")abs(PQ)=sqrt((r/sqrt(2))^2+(((sqrt(2)+1)/(sqrt(2)))r)^2)$
$XXX = (\sqrt{2 + \sqrt{2}}) r$ $color(white)("XXX")=(sqrt(2+sqrt(2)))r$

But we previously determined that
$XXX | P Q | = \sqrt{65}$ $color(white)("XXX")abs(PQ)=sqrt(65)$
So
$XXX r = \frac{\sqrt{65}}{\sqrt{2 + \sqrt{2}}}$ $color(white)("XXX")r=sqrt(65)/(sqrt(2+sqrt(2))$

The shortest arc length of an arc with radius $r$ $color(green)(r)$ and an angle of $\frac{3 π}{4}$ $(3pi)/4$ is $\frac{3}{4} r$ $3/4r$

$\Rightarrow$ $rArr$ shortest arc length is $\frac{3}{4} \times \frac{\sqrt{65}}{\sqrt{2 + \sqrt{2}}}$ $3/4xxsqrt(65)/(sqrt(2+sqrt(2)))$

$XXX = \frac{3 \sqrt{65}}{4 \sqrt{2 + \sqrt{2}}}$ $color(white)("XXX")=(3sqrt(65))/(4sqrt(2+sqrt(2)))$

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Points $(7, 1)$ $(7 ,1 )$ and $(8, 9)$ $(8 ,9 )$ are $\frac{3 π}{4}$ $(3 pi)/4$ radians apart on a circle. What is the shortest arc length between the points?

1 Answer

Explanation:

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Points (7,1)(7 ,1 ) and (8,9)(8 ,9 ) are 3π4(3 pi)/4 radians apart on a circle. What is the shortest arc length between the points?

1 Answer

Explanation:

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Points $(7, 1)$ $(7 ,1 )$ and $(8, 9)$ $(8 ,9 )$ are $\frac{3 π}{4}$ $(3 pi)/4$ radians apart on a circle. What is the shortest arc length between the points?