Using vectors prove that a quadrilateral is a rectangle if and
only if its diagonals are congruent and bisect each other.
Answers
Answer:
Let ABCD be a quadrilateral.
This quadrilateral is square.
So,
∣
AB
∣ = ∣
BC
∣ = ∣
CD
∣ = ∣
DA
∣
Now,
∣
CD
∣ = ∣
DC
∣
∣
DA
∣ = ∣
AD
∣
The diagonal are
AC
and
DB
It is square. Then the diagonals bisect each other.
We have to prove that : -
AC
⊥
DB
from square law of addition
We know that
AC
=
AB
+
AD
and
DB
=
AB
−
AD
to prove that : -
AC
⊥
DB
⇒
AC
∣.
DB
=0
So,
AC
.
DB
= (
AB
+
AD
).(
AB
−
AD
)
We know that,
(
a
+
b
).(
a
−
b
) = ∣
a
∣
2
- ∣
b
∣
2
⇒
AC
.
DB
= ∣
AC
∣
2
−∣
AD
∣
2
Since, ABCD is a square.
So,
⇒∣
AB
∣ = ∣
AD
∣
⇒
AC
∣.
DB
= 0
⇒
AC
⊥
DB
Hence, Proved.
Explanation: