Problems
01

POINT
Page
1
( 1 )
Show
that
P ( a,
b + c ),
Q ( b,
c + a )
and
R ( c,
a + b )
are
collinear.
( 2 )
Prove
that
the
two
lines
joining
the
midpoints
of
the
pairs
of
opposite
sides
and
the
line
joining
the
midpoints
of
the
diagonals
of
a
quadrilateral
are
concurrent.
( 3 )
If
( 2,
3 ),
( 4,
5 )
and
( a,
2 )
are
the
vertices
of
a
right
triangle,
find
a.
[ Ans:
3, 7
]
( 4 )
Find
the
circumcentre
of
the
triangle
with
vertices
(

1,
1 ),
( 0,

4 )
and
(

1,

5 )
and
deduce
that
the
circumcentre
of
the
triangle
whose
vertices
are
( 2,
3 ),
( 3,

2 )
and
( 2,

3 )
is
the
origin.
[ Ans:
(

3,

2 ) ]
( 5 )
For
which
value
of
a
would
the
area
of
a
triangle
with
vertices
( 5,
a ),
( 2,
5 )
and
( 2,
3 )
be
3
units ?
[ Ans:
For
any
a
∈
R ]
( 6 )
Find
the
area
of
the
triangle
whose
vertices
are
( l
2
,
2 l ),
( m
2
,
2 m )
and
(n
2
,
2 n )
if
l
≠
m
≠
n.
[ Ans:
l
( l

m ) ( m

n ) ( n

l )
l
]
( 7 )
Find
the
area
of
the
triangle
whose
vertices
are
( 5,
3 ),
( 4,
5 )
and
( 3,
1 )
and
show
that
the
triangle
whose
vertices
are
(

2,
2 ),
(

3,
4 )
and
(

4,
0 )
has
the
same
area.
[ Ans:
3
units ]
( 8 )
Find
the
area
of
the
triangle
with
vertices
( 5,
3 ),
( 4,
5 )
and
( 3,
1 )
by
shifting
the
origin
at
( 5,
3 ).
[ Ans:
3
units ]
( 9 )
Prove
that
the
midpoint
of
the
segment
joining
the
two
points
dividing
AB
from
A
in
the
ratios
m : n
and
n : m
is
the
midpoint
of
AB
.
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Problems
01

POINT
Page
2
( 10 )
If
P ( 1,
2 )
and
Q ( 5,
6 )
divide
AB
from
A
in
the
ratios
2
:
1
and

2
:
1,
find
the
coordinates
of
A
and
B.
[ Ans:
A (

1,
0 ),
B ( 2,
3 ) ]
( 11)
If
( 3,
2 ),
( 4,
5 )
and
( 2,
3 )
are three
of
the
four
vertices
of
a
parallelogram,
find
the
coordinates
of
the
fourth
vertex.
[ Ans:
( 5,
4 ),
( 3,
6 ),
( 1,
0 ) ]
( 12 )
Show
that
the
points
( 2,
3 ),
( 4,
5 )
and
( 3,
2 )
can
be
the
vertices
of
a
rectangle
and
find
the
coordinates
of
the
fourth
vertex.
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 Spring '06
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