# if two intersecting chords of a circle make equal angles with the diameter passing through their point of intersection , prove that the chords are equal.

• 5
The given information can be represented diagrammatically as: In this figure AB and CD are two chords intersect at point P. O is the centre of the circle. XY is the diameter of the circle passes through point P.
Given that ∠OPM = ∠OPN
Let OM⊥AB and ON⊥CD
In ∆OMP and ∆ONP
∠OMP = ∠ONP (Each 90°)
∠OPM = ∠OPN (Given)
OP = OP (Common)
∴ ∆OMP ≅ ∆ONP (AAS congruence criterion)
⇒ OM = ON (C.P.C.T)

Since the chords AB and CD are equidistant from the centre of the circle so, the chords AB and CD are equal
∴AB = CD
• 71
The given information can be represented diagrammatically as: In this figure AB and CD are two chords intersect at point P. O is the centre of the circle. XY is the diameter of the circle passes through point P.
Given that ∠OPM = ∠OPN
Let OM⊥AB and ON⊥CD
In ∆OMP and ∆ONP
∠OMP = ∠ONP(Each 90°)
∠OPM = ∠OPN(Given)
OP = OP(Common)
∴ ∆OMP ≅ ∆ONP(AAS congruence criterion)
⇒ OM = ON(C.P.C.T)
Since the chords AB and CD are equidistant from the centre of the circle so, the chords AB and CD are equal
∴AB = CD
• 17
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