Coordinate Geometry|| Chapter – 7|| Class 10th (For Hindi medium)
Introduction to Coordinate Geometry
Basics of Coordinate Geometry
Points on a Cartesian Plane
A pair of numbers locate points on a plane called the coordinates. The distance of a point from the y-axis is known as abscissa or x-coordinate. The distance of a point from the x-axis is called ordinates or y-coordinate.
Representation of (x, y) on the cartesian plane
Distance between Two Points on the Same Coordinate Axes
The distance between two points which are on the same axis (x-axis or y-axis), is given by the difference between their ordinates if they are on the y-axis, else by the difference between their abscissa if they are on the x-axis.
Distance AB = 6 – (-2) = 8 units
Distance CD = 4 – (-8) = 12 units
Distance between Two Points Using Pythagoras Theorem
Finding distance between 2 points using
Let P(x1, y1) and Q(x2, y2) be any two points on the cartesian plane.
Draw lines parallel to the axes through P and Q to meet at T.
ΔPTQ is right-angled at T.
By Pythagoras Theorem,
PQ2 = PT2 + QT2
= (x2 – x1)2 + (y2 – y1)2
PQ = √[x2 – x1)2 + (y2 – y1)2]
Distance between any two points (x1, y1) and (x2, y2) is given by
d = √[x2 – x1)2+(y2 – y1)2]
Where d is the distance between the points (x1,y1) and (x2,y2).
If the point P(x, y) divides the line segment joining A(x1, y1) and B(x2, y2) internally in the ratio m:n, then, the coordinates of P are given by the section formula as:
Finding ratio given the points
To find the ratio in which a given point P(x, y) divides the line segment joining A(x1, y1) and B(x2, y2),
- Assume that the ratio is k : 1
- Substitute the ratio in the section formula for any of the coordinates to get the value of k.
When x1, x2 and x are known, k can be calculated. The same can be calculated from the y- coordinate also.
The midpoint of any line segment divides it in the ratio 1 : 1.
The coordinates of the midpoint(P) of line segment joining A(x1, y1) and B(x2, y2) is given by
Points of Trisection
To find the points of trisection P and Q which divides the line segment joining
A(x1, y1) and B(x2, y2) into three equal parts:
i) AP : PB = 1 : 2
ii) AQ : QB = 2 : 1
Centroid of a triangle
If A(x1, y1),B(x2, y2) and C(x3, y3) are the vertices of a ΔABC, then the coordinates of its centroid(P) is given by
Area from Coordinates
Area of a triangle given its vertices
If A(x1, y1),B(x2, y2) and C(x3, y3) are the vertices of a Δ ABC, then its area is given by
A = 1/2[x1(y2 − y3) + x2(y3 − y1) + x3(y1 − y2)]
Where A is the area of the Δ ABC.
If three points A, B and C are collinear and B lies between A and C, then,
- AB + BC = AC. AB, BC, and AC can be calculated using the distance formula.
- The ratio in which B divides AC, calculated using section formula for both the x and y coordinates separately will be equal.
- Area of a triangle formed by three collinear points is zero.