From the center of the circle O to the chord AB equal to 20 cm, a perpendicular OС is drawn.

From the center of the circle O to the chord AB equal to 20 cm, a perpendicular OС is drawn. Find its length if the angle OBA = 45 °

We construct the radii OA and OB of the circle to the ends of the chord AB, then the triangle AOB is isosceles, and therefore the angle OBA = OAB = 450.

Then the angle AOB = (180 – OAB – OBA) = (180 – 45 – 45) = 900.

Triangle AOB is rectangular and isosceles.

The OS segment is the height, median and bisector of the triangle AOB, then AC = BC = AB / 2 = 20/2 = 10 cm.

The AOC triangle is rectangular and isosceles, since the AOC angle = COA = 450.

Then OC = AC = 10 cm.

Answer: The length of the OS segment is 10 cm.