Question

Equal chords AB and CD intersect each other at Q at right angle. P and R are mid points of AB and CD respectively. Show that OPQR is a square.

Here's the figure:http://goo.gl/ptW50

Answer 1

Hi friend ✋✋✋✋
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Your answer
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AB and CD are equal cords of a circle.

P and R are mid points of AB and CD respectively.

To prove : - OPQR is a square.

Now ,
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OP is a line dream from the centre of the circle on AB . Also, P is the mid point of AB. This, OP bisects AB .

we know that, any line from the centre of a circle on any chord which bisects it is perpendicular on it.

Then,
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Angle OPQ = 90°

Similarly , OQ bisects CD and is perpendicular to it.

So, Angle ORQ = 90°

Also,
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Equal cords are always equidistant from the centre.

So,
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OP = OQ

Thus, OPQR is a square.

Hence, proved.

HOPE IT HELPS

Answer 2

Step-by-step explanation:

Given : AB and CD are equal chords

intersecting at 90°

Construction : Join OQ

To prove : OPQR is a square

Proof : Since P and R are the mid-point of AB

and CD respectively

∴ ∠OPB = ∠ORD = 90°

∴ ∠OPQ = ∠ORQ = 90° 1

Since equal chords on a circle are equidistant

from the centre.

∴   OP = OR

Thus in DOPQ and DORQ, we have

OP = OR

∠OPQ = ∠ORQ 1

and OQ = OQ

∴ DOPQ ≅ DORQ [By R.H.S Congruence] 1

Thus in quadrilateral OPQR,

We have

OP = OR, PQ = RQ