0580

Circle Theorems

Geometry

The angle in a semicircle is 90°

  • Where a triangle has the diameter as one side and its third corner on the circumference, the angle at that corner is a right angle
  • To spot it, check that one side runs through the centre and that every corner lies on the circumference
  • Whichever corner faces the diameter is the one holding the right angle
A circle with centre O and a triangle drawn inside it: one side is a diameter running through O, the third corner sits on the circumference, and the angle at that corner is marked with a small square as a right angle
Source: Angle in a semicircle by Save My Exams
  • It is the special case of the centre theorem, since the straight angle of 180° at the centre halves to 90°
Worked example

Using the semicircle and the triangle

The diameter LN is drawn in a circle, and M is a third point on the circumference. Angle MLN is 28°. Find angle MNL, giving reasons.

A circle with a horizontal diameter drawn from L on the left to N on the right, and a third point M on the circumference above it. Triangle LMN is drawn, and the angle at L between the diameter and LM is marked 28 degrees. The angle at M is not labelled. The figure is marked NOT TO SCALE.

Solution:

  • LN is a diameter and M lies on the circumference, so angle LMN = 90°
  • The reason is that the angle in a semicircle is 90°
  • The three angles of triangle LMN add to 180°
  • Angle MNL = 180 − 90 − 28 = 62°
  • The second reason is that the angles in a triangle add to 180°

A tangent meets a radius at 90°

  • A tangent touches the circle at exactly one point, and the radius drawn to that point is perpendicular to it
  • Draw the radius in yourself whenever a tangent appears, because the right angle is usually what unlocks the question
  • Mark schemes accept "tangent and radius are perpendicular" as well as the ninety-degree form, and allow diameter in place of radius
A circle with centre O and a straight tangent touching it at one point, with the radius drawn out to that point and a small square marking the right angle between the two
Source: Circles & chords by Save My Exams