The chord MN divides a circle of radius 8 into two unequal arcs. It is visible from any point

| September 18, 2021 | education

The chord MN divides a circle of radius 8 into two unequal arcs. It is visible from any point on the smaller arc at an angle of 120 degrees. How long is the chord MN?

The degree measure of the larger arc MKN is equal to two degrees of the inscribed angle MAN.

Arc MKN = 2 * MAN = 2 * 120 = 240. Then the degree measure of the arc MAN = 360 – 240 = 120, and therefore the central angle MON = 120.

The height OH of the triangle MON is also its bisector and median, then HM = HN = MN / 2.

In the MANO quadrangle, the opposite angles are pairwise equal, and the diagonals OA and MN intersect at right angles, then MANO is a rhombus, and then AH = OH = OA / 2 = 8/2 = 4 cm.

In the right-angled triangle AMN, we determine the length of the leg MN.

tgMAH = MH / AH.

МН = АН * tg60 = 4 * √3 cm, then MN = 2 * МН = 8 * √3 cm.

Answer: The chord length is 8 * √3 cm.



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