##### 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 Formula**

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**Pythagoras Theorem

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 Formula**

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).

**Section Formula**

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:

P(x, y)=(mx2+nx1m+n,my2+ny1m+n)

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.

x=kx2+x1k+1

When x1, x2 and x are known, k can be calculated. The same can be calculated from the y- coordinate also.

**MidPoint**

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

p(x, y)=(x1+x22,y1+y22)

**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**

P=(x2+2×13,y2+2y13)

ii) **AQ : QB = 2 : 1**

Q=(2×2+x13,2y2+y13)

**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

p(x, y)=(x1+x2+x33,y1+y2+y33)

**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.

**Collinearity Condition**

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.