In the diagram, the tangent MN makes an angle of 55o with the chord PS. IF O is the centre of the circle, find < RPS

In the diagram, the tangent MN makes an angle of 55^o with the chord PS. IF O is the centre of the circle, find < RPS

Answer Details

Since MN is tangent to the circle, it must be perpendicular to the radius OP drawn to the point of contact. Therefore, angle MOP is 90 degrees. Let x be the measure of angle RPS. Then, angle RPO is also x degrees because RP is a radius and angles at the center are twice those at the circumference. Since MN is tangent to the circle, angle MPN is also 90 degrees. Since angle MPN and angle NPS form a linear pair (as they add up to 180 degrees) and angle MPN is 90 degrees, we have that angle NPS is 90 - x degrees. Finally, since angle MNQ and angle NPS are alternate angles, they are equal. Thus, angle MNQ is also 90 - x degrees. Now, we have a triangle MPN with angles 90, 90 - x, and 55 degrees. The angles of a triangle add up to 180 degrees, so we can write: 90 + (90 - x) + 55 = 180 Simplifying this equation gives: x = 35 Therefore, < RPS = 35 degrees.