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

| August 15, 2021 | education

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

From point O, the center of the circle, draw segments OA and OB equal to the radius of the circle.

The central angle AOB is equal to the degree measure of the arc that contracts the chord AB. Angle AOB = 120.

The AOB triangle is isosceles, since ОА = ОВ = R.

Let’s draw the height OH, which is also the median and bisector of the triangle AOB, then AH = BH = AB / 2 = 12 * √3 / 2 = 6 * √3 cm.

In the right-angled triangle AOН, the angle AOH = AOB / 2 = 120/2 = 600, then OA = R = AH / Sin600 = (6 * √3) / (√3 / 2) = 12 cm.

Let us determine the length of the arc AB by the formula: L = n * R * α / 1800, where R is the radius of the circle, and α is the angle between the radii.

L = n * 12 * 120/180 = 8 * n cm.

Let us determine the area of ​​the OAB sector.

Soav = n * R2 * α / 360 = n * 144 * 120/360 = 48 * n cm2.

Answer: The length of the arc is 8 * n cm, the area of ​​the sector is 48 * n cm2.



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