Coordinate Geometry

MATHEMATICS Sample test papers with hints Class-X (CBSE) MATHEMATICS

Coordinate Geometry Q.1 Prove that the points A(0,1), B(1,4), C(4,3), and D(3,0) are the vertices of a square. Q.2 Prove that the points A(2,-2), B(14,10), C(11,13), and D(-1,1) are the vertices of a rectangle. Q.3 Prove that the points (0,0), (5,5), (-5,5) are the vertices of a isosceles triangle. Q.4 Prove that the points (-2,5), (3,-4), (7,10) are the vertices of a right triangle. Q.5 Prove that the points (4,-1), (5,3), (2,5), and (1,1) are the vertices of a parallelogram. Q.7 Prove that the points (1,-2), (2,3), (-3,2), and (-4,3) are the vertices of a rhombus.. Q.8 Prove that the points (1,1), (-1,-1), (-

3, 3

), are the vertices of a equilateral

Q.9 Prove distance formula Q.10 prove section formula Q.11Prove that the coordinate of the centroid of a ∆ ABC with vertices A(x1,y1), B(x2,y2), C(x3,y3)are

given by

 x1 + x 2 + x 3 y1 + y 2 + y 3  ,   3 3  

∆

Q.12 Find the point on x-axis which is equidistant from the points (-2,5) and (2,3).[Ans.-2,0] Q.13 Find the point on y- axis which is equidistant from the points (-5,-2) and (3,2) [Ans.0,-2] Q.13 Find the value of y if the distance between the points (x,2) and (3,4) be 8 units. [Ans.3 or 9] −3

Q.14 Find the point which is three-fourth of the way from (3,1) to (-2,5). [Ans 4 ,4 ]. Q.15 Find the ratio in which the line segment joining the points (1,-3) and (4,5) is divided by the x-axis. [Ans.3:5] Q.16 Find the ratio in which the line segment joining the points (-2,4) and (7,3) is divided by the y-axis. [Ans.2:7] Q.17 Find the ratio in which the line 3x+y-9=0 divides the line segment joining the points (1,3) and (2,7). [Ans. 3:4] Q.18 Find the coordinates of a point whose distance from the point (3,5) is 5 units and that from (0,1) is 10 units. [Ans.6,9] Q.19 for what value m, the points (4,3), (m,1) and (1,9) are collinear? [Ans. m=5] Q.20 If the point P(x,y) is equidistant from the points A(5,1) and B(-1,5), prove that 3x=2y.

BASIC CONCEPTS Coordinate Geometry

The line segment OM in the units of scale chosen is called the x- coordinate or abscissa of point P. The line segment ON in the units of scale chosen is called the y-coordinate or ordinate of point P.

NOTE: The coordinates of the origin are taken as (0, 0) The coordinates of any point on x-axis are of the form (x, 0) The coordinates of any point on y-axis are of the form (0, y) Hence if the abscissa of a point is zero, it would lie somewhere on the y-axis. If the ordinate of a point is zero, it would lie somewhere on the x-axis. If the coordinate of a point p are (x, y) we shall frequently refer to it as P(x,y) Distance formula The distance b/w two points P(x1, y1) and Q(x2, y2) is given by PQ = Proof:

( x 2 −x1 ) 2 + ( y 2 − y1 ) 2

In order to prove that a given fig is a Square, prove that the four sides are equal and the diagonals are also equal. Rhombus, prove that the four sides are equal. Rectangle, prove that opposite sides are equal and the diagonals are also equal. Parallelogram, prove that the opposite sides are equal. Or Diagonals of a parallelogram bisect each other. Parallelogram but not a rectangle, prove that its opposite sides are equal but the diagonals are not equal Rhombus but not a square prove that its all sides are equal but the diagonals are not equal For three pints to be collinear prove that the sum of the distances b/w two pairs of points is equal to the third pair of points.

Section Formula:

 m1 x 2 + m 2 x 1 m 1 y 2 + m 2 y 1    , m1 + m 2   m1 + m 2

Proof:

Centroid of the triangle with vertices (x1, y1) (x2 ,y2) and (x3,y3) is given by  x 1 +x 2 + x 3 y1+ y 2 + y 3  ,  3 3 

   

Coordinate of mid point  x1 + x 2 y 1 + y 2  ,   2   2