Geometry and measuresGCSE Higher

A tangent is perpendicular to the radius at the point of contact

Recognise the theorem, follow two examples, then try five questions.

  • The radius must end at the point where the tangent touches the circle.

The theorem

tangent is perpendicular to the radius at the point of contact.

Example 1

With numerical values

is tangent at and is the centre. Find the angle .

The tangent is perpendicular to the radius.

Example 2

With algebra

is tangent at and is the centre. Find . Each labelled expression is an angle in degrees.

tangent and its radius meet at 90 degrees.Collect like terms.
Substitute the value of .
Labelled angles
tangent and its radius meet at 90 degrees.
Collect like terms.
Substitute the value of .
Labelled angles

Your turn

Question 1

Find angle .

Check answer
The tangent is perpendicular to the radius.
Question 2

Find the value of .

Check answer
tangent and its radius meet at 90 degrees.Collect like terms.
Substitute the value of .
Labelled angles
tangent and its radius meet at 90 degrees.
Collect like terms.
Substitute the value of .
Labelled angles
Question 3

Find angle .

Check answer
The tangent is perpendicular to the radius.
Question 4

Find the value of .

Check answer
tangent and its radius meet at 90 degrees.Collect like terms.
Substitute the value of .
Labelled angles
tangent and its radius meet at 90 degrees.
Collect like terms.
Substitute the value of .
Labelled angles
Question 5

Find angle .

Check answer
The tangent is perpendicular to the radius.
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