The chord of the circle is 12√3 and contracts the arc at 120 degrees. Find the arc length

| January 4, 2021 | education

The chord of the circle is 12√3 and contracts the arc at 120 degrees. Find the arc length and the area of the remaining sector.

The AOB triangle is isosceles, since ОА = ОВ = R.
Since the chord AB contracts the arc 1200, the central angle AOB is also equal to 1200.
We apply the cosine theorem for a triangle and determine the radius of the circle.
AB ^ 2 = OA ^ 2 + OB ^ 2 – 2 * OA * OB * Cos120.
AB ^ 2 = R ^ 2 + R ^ 2 – 2 * R * R * Cos120.
432 = 2 * R ^ 2 – 2 * R ^ 2 * (-1/2).
3 * R ^ 2 = 432.
R ^ 2 = 432/3 = 144.
R = 12 cm.
Let us determine the length of the smaller arc AB.
L = π * R * 120/180 = π * 12 * 120/180 = 8 * π cm.
Let’s define the area of the OAB sector.
Ssec = π * R ^ 2 * 120/360 = π * 144 * 120/360 = 432 * π cm2.
Answer: The length of the arc is 8 * π cm, the area of the sector is 432 * π cm2.



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