What angle is formed between a radius and a tangent?

**90°** (right angle) — a radius and a tangent meet at right angles.

What is the relationship between two tangents from the same external point?

They are **equal in length** — tangents from an external point are equal.

What does the perpendicular bisector of a chord do?

It **passes through the centre** of the circle — and a radius through the midpoint bisects the chord at right angles.

What shape do two tangents from the same point form with the circle?

A **kite** — with a vertical line of symmetry and two right angles where the tangents meet the radii.

In the worked example, why is angle TSO = 90°?

Because **a radius and a tangent meet at right angles** — OS is a radius and ST is a tangent at point S.

What triangles can you use when you have a chord and two radii?

**Isosceles triangles** — the two radii are equal, so the triangle has two equal sides and two equal base angles.

How do you find chord length when given radius and angle?

Draw the perpendicular from centre to chord (creates right angle), use **SOHCAHTOA** on the right triangle to find half the chord, then double it.

Theorems with Chords & Tangents | Pearson Edexcel IGCSE Maths

Q: What is a chord?

A chord is any **straight line** in a circle that joins **any two points** on the circumference.

Q: What is a tangent?

A tangent is a straight line **outside** the circle that touches its circumference at **exactly one point**.

Q: What exam phrase can you use for the chord-perpendicular theorem?

Either '**A radius bisects a chord at right angles**' or '**The perpendicular bisector of a chord passes through the centre**'.

Theorems with Chords & Tangents

Summary

This module covers three key circle theorems: (1) The perpendicular bisector of a chord passes through the centre — a radius bisects a chord at right angles. (2) A radius and a tangent meet at right angles. (3) Tangents from an external point are equal in length. These theorems are essential for solving problems involving chords, tangents, and finding lengths and angles in circle geometry.

1. Chords: Definition and Properties

A chord is any straight line in a circle that joins any two points on the circumference.

Chords of equal length are equidistant (the same distance) from the centre.

The perpendicular from centre O to chord PQ passes through midpoint M and meets PQ at 90°.

2. Theorem: Perpendicular Bisector of a Chord

Circle Theorem: The perpendicular bisector of a chord passes through the centre.

If a line through the centre (radius or diameter) goes through the midpoint of a chord, it will bisect (cut in half) that chord at right angles.

To spot: look for a radius intersecting a chord, or a radius that could bisect a chord.

Exam phrases: 'A radius bisects a chord at right angles' OR 'The perpendicular bisector of a chord passes through the centre'.

3. Tangents: Definition and Properties

A tangent to a circle is a straight line outside the circle that touches its circumference at exactly one point.

The radius OT is perpendicular to the tangent at point T.

4. Theorem: Radius and Tangent Meet at Right Angles

5. Theorem: Tangents from an External Point

6. Worked Example: Tangent Quadrilateral

7. Examiner Tips

Theorems with Chords & Tangents

Summary

1. Chords: Definition and Properties

A chord is any straight line in a circle that joins any two points on the circumference.

Chords of equal length are equidistant (the same distance) from the centre.

The perpendicular from centre O to chord PQ passes through midpoint M and meets PQ at 90°.

2. Theorem: Perpendicular Bisector of a Chord

Circle Theorem: The perpendicular bisector of a chord passes through the centre.

If a line through the centre (radius or diameter) goes through the midpoint of a chord, it will bisect (cut in half) that chord at right angles.

To spot: look for a radius intersecting a chord, or a radius that could bisect a chord.

Exam phrases: 'A radius bisects a chord at right angles' OR 'The perpendicular bisector of a chord passes through the centre'.

3. Tangents: Definition and Properties

A tangent to a circle is a straight line outside the circle that touches its circumference at exactly one point.

The radius OT is perpendicular to the tangent at point T.

4. Theorem: Radius and Tangent Meet at Right Angles

Circle Theorem: A radius and a tangent meet at right angles.

If a radius and a tangent meet at a point on the circumference, the angle between them is 90° — they are perpendicular.

Exam phrase: A radius and a tangent meet at right angles.

5. Theorem: Tangents from an External Point

Circle Theorem: Tangents from an external point are equal in length.

Two tangents from the same external point are equal in length.

This forms a kite with a vertical line of symmetry, from two congruent triangles back-to-back.

The kite has two right angles where the tangents meet the radii. Use Pythagoras and SOHCAHTOA on these triangles.

6. Worked Example: Tangent Quadrilateral

Problem: Find θ. ST and RT are tangents to the circle. Angle TSO = angle TRO = 90° (radius and tangent meet at right angles). Angle RTS = 25° (vertically opposite angles).

Angles in a quadrilateral sum to 360°: θ + 90 + 90 + 25 = 360 → θ + 205 = 360 → θ = 155°.

7. Examiner Tips

Look out for isosceles triangles formed by a chord and two radii.

When you see tangents, draw on the radius at right angles — it often helps.

Use SOHCAHTOA when you have right triangles with known angles.