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Q.

A quadrilateral whose diagonals are equal and bisect each other at right angles is called a:


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a

Rhombus

b

Square

c

Rectangle

d

Trapezium

answer is B.

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Detailed Solution

Concept- We’ll use the given property of quadrilateral and prove any unique property of quadrilateral by using the triangle similarity property, congruence rule and by its congruence property.
Let’s assume a quadrilateral ABCD whose diagonals are AC and BD intersecting each other at O.
Question Image       Diagonals AC and BD are equal and bisect each other at 90° .
So,
AC= BD
OB= OD
OA= OC
and
∠AOB=∠BOC=∠COD=∠AOD= 90°
In △AOB and △COD
AO= CO (Diagonals bisect each other)
OB= OD(Diagonals bisect each other)
∠AOB=∠COD= 90°(Vertically opposite angles)
From the SAS (side angle side) congruence rule, △AOB≅△COD
So, by the property that is referred as CPCT (Corresponding parts of congruent triangles)  AB=CD
∠OAB=∠OCD
∠OAB=∠OCD are the alternate interior angles of line AB and CD and alternate angles are equal when the lines are parallel to each other.
So, AB∥CD
From the above equations,
ABCD is a parallelogram.
In △AOD and △COD
AO= CO(Diagonals bisect each other)
OD= OD(Common)
∠AOD=∠COD= 90°
From the SAS (side angle side) congruence rule, △AOD≅△COD
So, via CPCT
AD=DC
However AD=BC and AB=CD due to the opposite sides of parallelogram ABCD.
So, AB=BC=CD=DA
Therefore, all the sides of the quadrilateral are equal to each other.
In △ADC and △BCD
AD= BC(proved)
AC= BD(Given)
CD=CD(Common)
From the SSS (side side side) congruence rule, △ADC≅△BCD
Again, through CPCT
∠ADC=∠BCD
The ∠ADC and ∠BCD are co- interior angles.
∠ADC+∠BCD=180°
∠ADC + ∠ADC=180°
2∠ADC=180°
∠ADC=90°
One of the interior angles of the quadrilateral is right angle.
So, we have obtained that ABCD is a parallelogram, AB=BC=CD=DA and one of the interior angles is right angle.
So, the quadrilateral ABCD is a square.
Hence, the correct option is 2) square.
 
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