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Prove that the circle drawn with any side of a rhombus as a diameter, passes through the point intersection of its diagonals.

Asked by sujaysarkar84 7th March 2015, 10:54 AM
Answered by Expert

Let PQRS be a rhombus with diagonals PR and QS which intersect at O.

We know that diagonals of rhombus intersect each other at right angles. Therefore

begin mathsize 12px style angle P O Q space equals space angle Q O R space equals space angle R O S space equals space angle S O P space equals space 90 degree end style

Here we have circles passing through PQ, QR, RS, SP as diametre and they pass through O.

For the circle with PQ as diameter, we have angle POQ inscribed in the semi-circle. Also angle inscribed in a semi-circle is a right angle.

Thus, the circle passes through O.

This is true for any circle constructed on any of the side of rhombus.

Therefore, circle with the side of rhombus as diameter passes through the point of intersection of its diagonals.

Answered by Expert 7th March 2015, 1:21 PM
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