Question

In a circle with center O. If OM is perpendicular to PQ, prove that PM = QM.

Answer 1

Given:
In the figure, O is centre of the circle and PQ is a  chord.
OM ⊥ PQ
To prove: PM = QM

Construction: Join Op and OQ
Proof:
In right triangles ΔOPM and ΔOQM,
OP = OQ        ....[radii of the same circle]
OM = OM      ....[common]
∴ By right angle-Hypotenuse-Side criterion of congruency,
ΔOPM ≅ΔOQM
The corresponding parts of the congruent triangles are congruent.
∴ PM = QM.