Question

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

Answer 1

Step-by-step explanation:

let we assume the quadrilateral ABCD and it's diagonals AC and BD

proof: quadrilateral is a rhombus

given: diagonals bisect each other at 90°

In ∆AOD and BOC

angle AOD=Angle BOC (vertically opposite angle)

AO=OC ( given)

DO=OB(Given)

so ∆AOD is congruent to∆BOC by SAS congruency

now AD=BC ( by c.p.c.t).......(1)

similarly, the ∆ AOB and ∆DOC get congruent and we get that the

AB=DC ........(2)

Now , in triangle ABC and ∆ADC

AC=CA(common)

AD=BC( proofed above)

AB=DC( proofed above)

so ∆ ABC and∆ADC are congruent and

AB=BC and AD = DC ( by c.p.c.t)..........(3)

similarly, the ∆ DAB and ∆DCB get congruent and we get

AD= AB and DC=CB ( by c.p.c.t)......(4)

on combining all the equations we get

AB = BC=CD=DA

all the sides are equal which is the property of a rhombus

hence proof

hope it helps you