Answer: pi
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Step-by-step explanation:
r = radius of the circle = 3
This is because segment IJ is 3 units.
Let's find the area of a full circle based on this radius.
A = pi*r^2
A = pi*3^2
A = 9pi
That's the area of a full circle with radius 3.
However, we only have a small pizza slice shaded. This slice has area of (1/2)pi or pi/2
If we divided the area of the slice over the full area, then we can figure out how many equal such slices make up the full circle.
(9pi)/(pi/2) = 9pi*(2/pi) = 18
This pizza has 18 slices total. Each slice has area pi/2 and it gives a total area of 18*(pi/2) = 9pi
The jump from pi/2 to 9pi is exactly "times 18" to represent the 18 slices.
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Since we now know that this pizza has 18 equal slices, we'll cut a full 360 degree rotation into 18 equal pieces
360/18 = 20
Therefore, the central angle KIJ is 20 degrees.
Plug this central angle, along with the radius, into the arc length formula
L = arc length
L = (circumference)*(angle in degrees)/360
L = (2*pi*r)*(20/360)
L = 2pi*9*(1/18)
L = pi
Arc KJ is exactly pi units long.
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Paraphrased another way:
The full distance around the circle is 2*pi*r = 2*pi*9 = 18pi units
However, we only want a slice of this perimeter. We divide by 18 because we found earlier that there are 18 equal slices.
So each slice has its curved portion, aka arc length, being (18pi)/18 = pi units