Circle theorems - Higher - EdexcelTangents - Higher

There are two circle theorems involving tangents.

1. The angle between a tangent and a radius is 90°.

2. Tangents which meet at the same point are equal in length.

Example

Calculate the angles EFG and FOG.

Triangle GEF is an isosceles triangle.

Angle FGE = angle EFG

The angle between the tangent and the radius is 90°.

Angle EFO = EGO = 90°

The shape FOGE is a quadrilateral. The angles in a quadrilateral add up to 360°.

Proof

The angle between the tangent and the radius is 90°.

Angle BCO = angle BAO = 90°

AO and OC are both radii of the circle.

Length AO = Length OC

Draw the line OB. It creates two triangles OCB and OAB. These share the length OB.

Triangles OCB and OAB are congruent because of the RHS rule.

The hypotenuse OB is common to both triangles.

Sides OC and OA are radii and therefore are equal.

The triangles are right-angled as the angle between the tangent and the radius is 90°.

Therefore triangles OAB and OCB are congruent.

So length CB = AB.