Tangents - Higher

There are two circle theorems involving tangents.

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

Circle with radius and tangent shown

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

Example

Calculate the angles EFG and FOG.

Circle with 2 identical tangents from point E at angle, 20degrees

Triangle GEF is an isosceles triangle.

Angle FGE = angle EFG

FGE = EFG = \frac{180 - 20}{2} = 80^\circ

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°.

Angle FOG = 360 - 90 - 90 - 20 = 160^\circ

Proof

Circle with 2 identical tangents from point B.

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

Circle with 2 identical tangents from point B plus triangles (AOB) and (COB)

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 SAS rule.

Two of the sides are the same length: OB = OB and OC = OA

One of the angles is equal in size: OCB = OAB

Congruent triangles are identical.

So length CB = AB.