X- coordinate(Abscissa) : It is the distance of a point from axis.
Y- coordinate(Ordinate) : It is the distance of a point from axis.
Distance formula: Given
The distance from to is
Section Formula:Given points if point which
divides the line segment internally in the ratio m:n, then coordinates
of points are
Mid point formula: If P is mid point of segment AB,where
Area of triangle : If are vertices
of triangle then area of triangle is given by
* To find
the area of a polygon we divide it in triangles and take numerical value of the
area of each of the triangles.
* The area of can also be computed by using the following steps:
STEP I: Write the
coordinates of the vertices and in three
columns as shown below and augment the coordinates of as fourth column.
STEP II:Draw broken parallel lines pointing down wards from left
to right and right to left.
the sum of the products of numbers at the ends of lines pointing
downwards from left to right and subtract from this sum, the sum
of the products of numbers at the ends of the lines pointing downward from
right to left i.e., compute
STEP IV: Find the absolute of
the number obtained in step III and take its half to obtain the area.
* Three points and are collinear if
Area of i.e.,
THEOREM 1: Prove that the
coordinates of the centroid of the triangle whose vertices are
Also deduce that the medians of a
triangle are concurrent.
[NCERT, CBSE 2004]
PROOF: Let and be the vertices
of whose medians are
AD, BE and respectively. So and are respectively
the mid-points of and .
Coordinates of are
Coordinates of a point dividing in the ratio are
The coordinates of E are
The coordinates of a point dividing BE in the ratio are
Similarly the coordinates of a point dividing CF in the ratio are
Thus, the point having coordinates is common to and and divides them
in the ratio .
Hence, medians of a triangle are concurrent and the coordinates of
the centroid are
THEOREM 2:The area of a
triangle, the coordinates of whose vertices are and is
PROOF: LET be a triangle whose
vertices are and . Draw ,
BM and perpendiculars from on the axis.
Clearly, and are all trapeziums.
We know that
Area of trapezium (Sum of parallel
sides)(Distance between them)
Area of = Area of
trapezium Area of trapezium
Area of trapezium Let denote the area
of . Then,