The sum of the degree measures of the angle ABC inscribed in the circle and the central angle AOC is equal to 90 °

| April 1, 2021 | education

The sum of the degree measures of the angle ABC inscribed in the circle and the central angle AOC is equal to 90 °. Calculate: a) the degree measures of the angles ABC and AOC; b) the distance from the center of the circle to the chord AC, if the radius of the circle is 8 cm.

The inscribed angle is half of the central angle resting on the same arc.

Let the inscribed angle ABC = X0, then the central angle AOC = 2 * X0.

By condition, ABC + AOC = 90.

X + 2 * X = 90.

X = ABC = 30, AOC = 60.

In the AOC triangle, OA = OC = R = 8 cm, and the AOC angle = 60, then the AOC triangle is equilateral. The OH segment is the height of an equilateral triangle, then OH = √3 * 8/2 = 4 * √3 cm.

Answer: Angle ABC = 30, angle AOC = 60, distance to the chord is 4 * √3 cm.



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