Let PQRS be a quadrilateral. If M and N are the
mid points of the sides PO and RS respectively
then prove that PS+QR=2MN.
Answers
Answer:
Let \overline{m}, \overline{n}, \overline{p}, \overline{q}, \overline{r}
m
,
n
,
p
,
q
,
r
and \overline{s}
s
be position vectors of M,N,P,Q,RM,N,P,Q,R and SS respectively.
MM and NN are midpoints of PQPQ and RSRS
\therefore \overline {m} = \dfrac {\overline {p} + \overline {q}}{2}∴
m
=
2
p
+
q
and
\overline {n} = \dfrac {\overline {r} + \overline {s}}{2}
n
=
2
r
+
s
\overline {PS} + \overline {QR} = \overline {s} - \overline {p} + \overline {r} - \overline {q}
PS
+
QR
=
s
−
p
+
r
−
q
= (\overline {r} + \overline {s}) - (\overline {p} + \overline {q})=(
r
+
s
)−(
p
+
q
)
= 2\overline {n} - 2\overline {m}=2
n
−2
m
= 2(\overline {n} - \overline {m})=2(
n
−
m
)
= 2\overline {MN}=2
MN
Step-by-step explanation: