Chord theorems

The line joining the mid-point of a chord to the centre of a circle is perpendicular to the chord. This can be proved using congruent triangles.

Image

AB is a chord to the circle centre O.

X is the mid-point of AB. In triangles OXA and OXB:

OA = OB (radii)

AX = XB ( is the mid-point)

OX is common

Therefore Triangles and are congruent (SSS)

Therefore Angle OXA = angle OXB = 90°

The video below highlights the rules you need to remember to work out circle theorems.