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 .