Question

The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord. Prove it.

Answer 1

hey user here is your answer:-

first of all it is given that the line is drawn through the centre of circle now if you join the centre with the terminal points of the cord then you will get two triangles now you had to prove congruent two triangles so you can prove it
as the radii are equal the common chord and it is also given the radii bisects the chord
so you get that your two Triangles are congruent after that you get that your two angles on the chord are also equal by cpct after that by applying linear pair theorem you can get your angle is 90 degree hope It helped

Answer 2

Statement : The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.

Given : A chord PQ of a circle C ( O, r ) and L is the mid - point of PQ.

To prove : OL ⊥ PQ

Construction : Joint OP and OQ.

PROOF :

In ∆OLP and ∆OLQ,

OP = OQ . (radii of same circle)
PL = QL . (given)
OL = OL . (common)

ΔOLP ≅ ΔOLQ . (SSS)

Also,

∠OLP + ∠OLQ = 180° (linear pair)
∠OLP = ∠OLQ = 90°

Hence, OL ⊥ PQ.