Question

11. ABCD is a parallelogram. What kind of
quadrilateral is it if :
(i) AC = BD and AC is perpendicular to
BD ?
AC is perpendicular to BD but is not equal
to it ?
(iii) AC = BD but AC is not perpendicular to
BD ?
(ii)​

Answer 1

Answer:

Given: In parallelogram ABCD, the diagonals AC &  BD are equal &  perpendicular to each other.

∠DOC=∠COB=∠BOA=∠AOD=90  

o

 

We know that diagonals of a parallelogram bisect each other.  

So OA=OC=OB=OD ---(1)

In △AOD&△AOB,

OA=OA  [ Common side ]  

∠BOA=∠AOD=90  

o

 [ Given ]  

OD=OA [ from (1) ]  

So △AOD≅△AOB [ By SAS Rule of Congruency ]  

Therefore AD=AB.

So AB=BC=CD=DA [ Since ABCD is a parallelogram ]  ---(2)

In △ADB & △BCA,

AB=AB [ Common side ]  

AD=BC [ from (2) ]

AC=BD [ given ]  

Therefore △ADB≅△BCA [ By SSS Rule of Congruency ]  

So ∠DAB=∠CBA

Now adjacent angles in a parallelogram are supplementary.

So ∠DAB+∠CBA=180  

o

 

⇒2∠DAB=2∠CBA=180  

o

 

⇒∠DAB=∠CBA=  

2

180  

o

 

​  

 

⇒∠DAB=∠CBA=90  

o

 

Similarly we can prove that ∠ADC=∠BCD=90  

o

 

So in the given parallelogram, all sides are equal & each interior angle is 90  

o

.

Therefore parallelogram ABCD in which diagonals are equal &  are perpendicular to each other is definitely a Square.  

Step-by-step explanation: