Circle Theorems

Circle Theorems Introduction

• Circle theorems deal with angles formed by lines drawn inside and around circles.
• Two key theorems: angle in a semicircle is 90°, and radius meets tangent at 90°.
• These theorems help solve problems involving circles, triangles, and tangents.

Angle in a Semicircle

• If a triangle has its vertices on the circumference and one side is a diameter, then the angle opposite the diameter is 90°.
• This is called 'the angle in a semicircle is 90°'.
• The right angle is always at the point on the circumference, not at the ends of the diameter.

In the diagram, AC is a diameter. Which angle is 90°?

• Angle ABC
• Angle BAC
• Angle ACB
• None of these

Tangent and Radius

• A tangent is a straight line that touches the circle at exactly one point.
• The radius drawn to the point of tangency is perpendicular to the tangent.
• This means the angle between the radius and the tangent is 90°.

The angle between a ____ and a tangent is ____ degrees.

Using the Theorems Together

• In problems, you may need to combine both theorems with triangle angle sum (180°).
• Always mark right angles on the diagram when you spot a diameter or a tangent.
• Look for isosceles triangles formed by two radii (equal sides).

Which of these are true?

• The angle in a semicircle is always 90°.
• A tangent touches the circle at two points.
• A radius and a tangent are perpendicular.
• The diameter is twice the radius.

Example: Angle in a Semicircle

• Points A, B, C on a circle with AC as diameter.
• Angle $ABC = 90^{\circ} (angle$ in semicircle).
• If angle $BAC = 53^{\circ}$, then angle $ACB = 180 - 90 - 53 = 37^{\circ}$.

In a circle with diameter AC, angle $BAC = 40^{\circ}$. What is angle ABC?

• 40°
• 50°
• 90°
• 140°

Example: Tangent and Radius

• P is a point on the circle, APB is a tangent at P, O is the centre.
• OP is a radius, so angle $OPB = 90^{\circ}$.
• If angle $APO = 53^{\circ}$, then angle $OPQ = 37^{\circ} (since OPB = 90^{\circ})$.

In the diagram, OP is a radius and APB is a tangent. Angle OPB = ____^{\circ}.

Isosceles Triangles in Circles

• Two radii from the centre to the circumference are equal, forming an isosceles triangle.
• Base angles of an isosceles triangle are equal.
• This fact is often used with circle theorems to find unknown angles.

Which statements about triangle OPQ (O centre, P and Q on circumference) are true?

• $OP = OQ$
• Angle $OPQ =$ angle OQP
• Triangle OPQ is always right-angled

Solving Problems Step by Step

• Step 1: Identify any diameters or tangents in the diagram.
• Step 2: Mark the 90° angles (angle in semicircle or radius-tangent).
• Step 3: Use triangle angle sum (180°) and isosceles triangle properties to find missing angles.

In a circle with centre O, AB is a diameter. C is on the circumference. Which is true?

• Angle $ACB = 90^{\circ}$
• Angle $OAC = 90^{\circ}$
• Angle $ABC = 90^{\circ}$
• Angle $AOC = 90^{\circ}$

Common Exam Tips

• As soon as you spot a diameter or tangent, mark the right angle on the diagram.
• Use keywords in explanations: 'angle in a semicircle is 90°' or 'radius and tangent meet at right angles'.
• Sometimes just marking the right angle earns a mark!

The reason angle $ABC = 90^{\circ}$ is because it is the angle in a ____.

Summary

• Two main circle theorems: angle in a semicircle $= 90^{\circ}$, radius ⟂ tangent.
• Combine with triangle angle sum and isosceles triangle properties.
• Practice identifying these in diagrams to solve problems quickly.

A tangent touches a circle at point T. The radius OT is drawn. What is angle OTA?

• 0°
• 45°
• 90°
• 180°