Question

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Q10. Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.

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Answer 1

Take quadrilateral ABCD , AC and BD are diagonals which intersect at O.

Take quadrilateral ABCD , AC and BD are diagonals which intersect at O.In △AOB and △AOD

Take quadrilateral ABCD , AC and BD are diagonals which intersect at O.In △AOB and △AODDO=OB ∣ O is the midpoint

Take quadrilateral ABCD , AC and BD are diagonals which intersect at O.In △AOB and △AODDO=OB ∣ O is the midpointAO=AO ∣ Common side

Take quadrilateral ABCD , AC and BD are diagonals which intersect at O.In △AOB and △AODDO=OB ∣ O is the midpointAO=AO ∣ Common side∠AOB=∠AOD ∣ Right angle

Take quadrilateral ABCD , AC and BD are diagonals which intersect at O.In △AOB and △AODDO=OB ∣ O is the midpointAO=AO ∣ Common side∠AOB=∠AOD ∣ Right angleSo, △AOB≅△AOD

Take quadrilateral ABCD , AC and BD are diagonals which intersect at O.In △AOB and △AODDO=OB ∣ O is the midpointAO=AO ∣ Common side∠AOB=∠AOD ∣ Right angleSo, △AOB≅△AODSo, AB=AD

Take quadrilateral ABCD , AC and BD are diagonals which intersect at O.In △AOB and △AODDO=OB ∣ O is the midpointAO=AO ∣ Common side∠AOB=∠AOD ∣ Right angleSo, △AOB≅△AODSo, AB=ADSimilarly, AB=BC=CD=AD can be proved which means that ABCD is a rhombus.

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