Mathematics
The given figure shows a circle with center O. P is mid-point of chord AB. Show that OP is perpendicular to AB.
Triangles
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Answer
Join OA and OB.
In △ OAP and △ OBP,
⇒ OP = OP (Common side)
⇒ OA = OB (Radius of same circle)
⇒ AP = PB (Since, P is mid-point of chord AB)
∴ △ OAP ≅ △ OBP (By S.S.S. axiom)
We know that,
Corresponding parts of congruent triangles are equal.
∴ ∠OPA = ∠OPB = x (let)
Since, AB is a straight line.
∴ ∠OPA + ∠OPB = 180°
⇒ x + x = 180°
⇒ 2x = 180°
⇒ x = = 90°.
Hence, proved that OP is perpendicular to chord AB.
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