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)
Answers
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: