The properties of a rectangle, rhombus and square are discussed here using figure.
Let ABCD be a rectangle whose diagonals AC and BD intersect at the point 0.
From ∆ ABC and ∆ BAD,
AB = BA (common)
∠ABC = ∠BAD (each equal to 90o)
BC = AD (opposite sides of a rectangle).
Therefore, ∆ ABC ≅ ∆ BAD (by SAS congruence)
⇒ AC = BD.
Hence, the diagonals of a rectangle are equal.
From ∆ OAB and ∆ OCD,
∠OAB = ∠OCD (alternate angles)
∠OBA = ∠ODC (alternate angles)
AB = CD (opposite sides of a rectangle)
Therefore, ∆OAB ≅ ∆ OCD. (by ASA congruence)
⇒ OA = OC and OB = OD.
This shows that the diagonals of a rectangle bisect each other.
Hence, the diagonals of a rectangle are equal and bisect each other.
Let ABCD be a rhombus whose diagonals AC and BD intersect at the point O.
We know that the diagonals of a parallelogram bisect each other.
Also, we know that every rhombus is a parallelogram.
So, the diagonals of a rhombus bisect each other.
Therefore, OA = OC and OB = OD
From ∆ COB and ∆ COD,
CB = CD (sides of a rhombus)
CO = CO (common).
OB = OD (proved)
Therefore, ∆ COB ≅ ∆ COD (by SSS congruence)
⇒ ∠COB = ∠COD
But, ∠COB + ∠COD = 2 right angles (linear pair)
Therefore, ∠COB = ∠COD = 1 right angle.
Hence, the diagonals of a rhombus bisect each other at right angles.
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We know that the diagonals of a rectangle are equal.
Also, we know that every square is a rectangle.
So, the diagonals of a square are equal.
Again, we know that the diagonals of a rhombus bisect each other at right angles. But, every square is a rhombus.
So, the diagonals of a square bisect each other at right angles.
Hence, the diagonals of a square are equal and bisect each other at right angles.
If the diagonals of a quadrilateral are equal then it is not necessarily a rectangle.
In the adjacent figure, ABCD is a quadrilateral in which diagonal AC = diagonal BD, but ABCD is not a rectangle.
If the diagonals of a quadrilateral intersect at right angles then it is not necessarily a rhombus.
Parallelogram
Properties of a Rectangle Rhombus and Square
Practice Test on Parallelogram
Parallelogram  Worksheet
8th Grade Math Practice
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