Question 5
Prove that the circle drawn with any side of a rhombus as diameter passes through the point of intersection of its diagonals.

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Solution

Let ABCD be a rhombus in which diagonals are intersecting at point O and a circle is drawn while taking side CD as its diameter.
We know that a diameter subtends $90^{\circ}$ on the arc
$∴ ∠ C O D = 90^{\circ}$
Also, in rhombus, the diagonals intersect each other at $90^{\circ}$ .
$∴ ∠ A O B = ∠ B O C = ∠ C O D = ∠ D O A = 90^{\circ}$

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Rhombus

Standard IX Mathematics

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Question 5 Prove that the circle drawn with any side of a rhombus as diameter passes through the point of intersection of its diagonals.

Let ABCD be a rhombus in which diagonals are intersecting at point O and a circle is drawn while taking side CD as its diameter. We know that a diameter subtends 90∘ on the arc ∴∠COD=90∘ Also, in rhombus, the diagonals intersect each other at 90∘. ∴∠AOB=∠BOC=∠COD=∠DOA=90∘

Question 5
Prove that the circle drawn with any side of a rhombus as diameter passes through the point of intersection of its diagonals.