IGCSE Maths / Circle theorems

Tangent and radius, and two tangents from a point

Edexcel International GCSE Maths (4MA1), chapter 17: Circle theorems, specification point 4.6c-f. Free lesson.

Read the revision notes · IGCSE Maths past papers

A tangent meets the radius at its point of contact at 90°, which lets you use Pythagoras. Two tangents from the same point outside a circle are equal, and together with the radii they form a kite.

Where the right angle is

A tangent touches a circle once, then leaves it. At the point where it touches, the tangent is perpendicular to the radius, so the angle between them is 90°. This is the key to spotting right-angled triangles in tangent questions. Draw the radius to the point where the tangent touches. The right angle is between the tangent and the radius, not between the tangent and a chord.

Finding an angle between two radii

Two tangents from P touch a circle at A and B. The radii OA and OB meet the tangents at 90°, making the four-sided shape OAPB a quadrilateral. Its interior angles add to 360°, so you can subtract the known angles to find the angle at the centre.

Using a tangent and radius with Pythagoras

A radius and a tangent form a right-angled triangle at the point of contact. In triangle OAP, OP is opposite the right angle, so it is the hypotenuse, the longest side. To find a shorter side, subtract the other shorter side squared from the hypotenuse squared, then take the square root. For the example, OA = 5 cm and OP = 13 cm.

Why two tangents from one point are equal

The triangles OAP and OBP share OP. They also have equal radii OA and OB, and each has a right angle. The triangles are therefore congruent, which means their tangent sides have equal lengths. The shape OAPB is a kite. PA = PB. The kite’s angles add to 360°, so this fact can also help you find an angle.

Finding the centre angle in a kite

The angle at the centre is the fourth angle in OAPB. Subtract the two right angles and the angle between the tangents from 360°.

More lessons in Circle theorems

• Parts of a circle and cyclic quadrilaterals
• The angle at the centre is twice the angle at the circumference
• The angle in a semicircle
• Angles in the same segment
• Opposite angles of a cyclic quadrilateral
• The alternate segment theorem
• Using the alternate segment theorem in a reasoning chain
• Reasoning chains with circle theorems
• Multi-step circle theorem problems
• Intersecting chords inside a circle
• Intersecting chords from a point outside the circle