Everything Maths Grade 12

CHAPTER 9. GEOMETRY 9.2

Proof:

�

A B

O

P

Consider a circle, with centre O. Draw a chord AB and draw a perpendicular line from the centreof
the circle to intersect thechord at point P.

The aim is to prove that AP = BP

1.�OAP and�OBP are right-angled triangles.

2. OA = OB as both of these are radii and OP is common to both triangles.

Apply the Theorem of Pythagoras to each triangle, to get:

OA^2 = OP^2 +AP^2

OB^2 = OP^2 +BP^2

However, OA = OB. So,

OP^2 +AP^2 = OP^2 +BP^2

∴ AP^2 = BP^2

and AP = BP

This means that OP bisects AB.

Theorem 2. The line drawn from the centre of a circle, that bisects a chord, is perpendicular to the
chord.

Proof:

�

A B

O

P

Consider a circle, withcentre O. Draw a chord AB and draw a line from the centre of the circle to
bisect the chord at point P.

The aim is to prove that OP⊥ AB

In�OAP and�OBP ,