A circle's center is at (4 ,2 )(4,2) and it passes through (6 ,7 )(6,7). What is the length of an arc covering (5pi ) /3 5π3 radians on the circle?

1 Answer
Apr 6, 2016

Arc length~~28.228.2 to 1 decimal place

Explanation:

Let the radius of the circle be rr
Let the length of arc be L_aLa

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")Determine the radius of the circle

=> r=sqrt((x_2-x_1)^2+(y_2-y_1)^2)r=(x2x1)2+(y2y1)2

=> r = sqrt((6-4)^2 +(7-2)^2)r=(64)2+(72)2

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

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

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

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

So the length of arc L_a=rxx (5pi)/3La=r×5π3

=> L_a=sqrt(29)xx (5pi)/3La=29×5π3

color(blue)(L_a~~28.2" to 1 decimal place")La28.2 to 1 decimal place

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

Circumference = piD = pixx2sqrt(29)πD=π×229

and we have 1 2/3" of "1/2 123 of 12 of the circumference

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