Geometry and measuresGCSE Higher

The two tangent segments from the same external point to a circle have equal lengths

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

  • Triangle is isosceles, so its base angles are equal.

The theorem

The two tangent segments from the same external point to a circle have equal lengths.

Example 1

With numerical values

and are tangents at and . Find the angle .

Equal tangents make triangle isosceles, so its base angles are equal.

Example 2

With algebra

and are tangents at and . Find . Each labelled expression is an angle in degrees.

Equal tangents make triangle isosceles, so its base angles are equal.Collect like terms.
Substitute the value of .
Labelled angles

Your turn

Question 1

Find angle .

Check answer
Equal tangents make triangle isosceles, so its base angles are equal.
Question 2

Find the value of .

Check answer
Equal tangents make triangle isosceles, so its base angles are equal.Collect like terms.
Substitute the value of .
Labelled angles
Question 3

Find angle .

Check answer
Equal tangents make triangle isosceles, so its base angles are equal.
Question 4

Find the value of .

Check answer
Equal tangents make triangle isosceles, so its base angles are equal.Collect like terms.
Substitute the value of .
Labelled angles
Question 5

Find angle .

Check answer
Equal tangents make triangle isosceles, so its base angles are equal.
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