The circle theorems

5) Tangents

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.

Angle FGE = angle EFG = (180 – 20) /2 = 80°

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

Angle EFO = angle EGO = 90°

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

Angle FOG = 360 – 90 – 90 – 20 = 160°


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. Two of the sides are the same length: OB = OB and OC = OA. One of the angles in each triangle is a right angle: OCB = OAB. Congruent triangles are identical.

So length CB = AB.


< Previous | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Next >>