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Question
The diagonals of a quadrilateral bisect each other at right angles. Show that the quadrilateral is a rhombus.
Solution
Since, the diagonals AC and BD of quadrilateral ABCD bisect each other at right angles.
∴ AC is the ⊥ bisector of line segment BD
⇒ A and C both are equidistant from B and D
⇒ AB = AD and CB = CD ...(i)
Also, BD is the ⊥ bisector of line segment AC
⇒ B and D both are equidistant from A and C
⇒ AB = BC and AD = DC ...(ii)
From (i) and (ii), we get
AB = BC = CD = AD
Thus, ABCD is a quadrilateral whose diagonals bisect each other at right angles and all four sides are equal.
Hence, ABCD is a rhombus.
Hence proved.
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