The length of the radius of a circle is 12. Find the area of a segment bounded by an arc with measure of 60 degree and the corresponding chord.
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The length of the radius of a circle is 12. Find the area of a segment bounded by an arc with measure of 60 degree and the corresponding chord.
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PAG TAMA PO
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Verified answer
To find the area of the segment, we first need to find the area of the sector that corresponds to the arc with a measure of 60 degrees.
The formula for the area of a sector is: (angle/360) * pi * r^2
where angle is the measure of the arc in degrees, pi is approximately 3.14, and r is the radius of the circle.
So in this case, the area of the sector is: (60/360) * 3.14 * 12^2 = 3.14 * 144/6 = 3.14 * 24 = 75.36 sq units
We can then subtract the area of the triangle formed by the center of the circle, the midpoint of the chord and the endpoints of the chord from the sector area to get the segment area.
The area of this triangle can be calculated using Heron's formula or (1/2)* b* h (base and height)
The base of the triangle is the length of the chord, which is (212sin(30)) = 12*sqrt(3)
The height of the triangle is the distance from the center of the circle to the midpoint of the chord. This distance is equal to the radius of the circle, which is 12.
so the area of the triangle is (1/2)*12sqrt(3)*12 = (1/2) * 144sqrt(3) = 72sqrt(3) sq units
Therefore the area of the segment is 75.36sq units - 72sqrt(3) sq units = 75.36 - 72sqrt(3) sq units.