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.- 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^2However, OA = OB. So,
OP^2 +AP^2 = OP^2 +BP^2
∴ AP^2 = BP^2
and AP = BPThis 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 ,