Question

A circle's center is at `(4 ,2 )` and it passes through `(6 ,7 )`. What is the length of an arc covering `(5pi ) /3` radians on the circle?

Answer 1

Arc length`~~28.2` to 1 decimal place

Explanation:

Let the radius of the circle be `r`
Let the length of arc be `L_a`

Distance from the circles centre to any point on its circumference is always the same.
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`color(blue)("Determine the radius of the circle")`

`=> r=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`

`=> r = sqrt((6-4)^2 +(7-2)^2)`

`=>color(blue)(r = sqrt(29))" "->` 29 is a prime number so can not be simplified

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`color(blue)("Determine the length of arc")`

`color(brown)("Important point:")`

`color(brown)("1 radian is such that its length of ark is the same as")` `color(brown)("the length of the radius.")`

So the length of arc `L_a=rxx (5pi)/3`

`=> L_a=sqrt(29)xx (5pi)/3`

`color(blue)(L_a~~28.2" to 1 decimal place")`

'~~~~~~~~~~~~~~~~~~~~~~~~~

`color(blue)("Check:")`

Circumference = `piD = pixx2sqrt(29)`

and we have `1 2/3" of "1/2` of the circumference

`=>L_a=1 2/3 xxpisqrt(29) =28.19...` Confirmed!