Question

Prove that the line joining the midpoint of chord and center of the circle is perpendicular to the chord?

Answer 1

Given :- (from image) .

• AB is a chord .
• C is mid - point of AB.
• O is centre of circle.

To Prove :-

• OC ⊥ AB.

Solution :-

in ∆ABO and ∆ACO , we have,

→ AB = AC (C is mid point of AB.)

→ OC = OC (common.)

→ OA = OB (Radius of circle.)

So,

→ ABO ≅ ∆ACO (By SSS congruence.)

Therefore,

→ ∠OCA = ∠OCB (By CPCT.)

Now,

→ ∠OCA + ∠OCB = 180° (AB is a straight line)

So,

→ 2∠OCA = 180°.

→ ∠OCA = 90° .

Therefore,

→ ∠OCA = ∠OCB = 90°

Hence,

→ OC ⊥ AB.

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