Chapter: Coordinate Geometry - Class 10

Coordinate geometry is a branch of mathematics that combines algebraic techniques with geometry. This chapter includes the formula to find the distance between the two points whose coordinates are given, the area and centroid of the triangle formed by the three given points and the section formula.

Coordinate Geometry

Coordinate geometry is a branch of mathematics that combines algebraic techniques with geometry. It is also known as analytic geometry.

Every point is represented by an ordered pair known as the coordinates of that point. An ordered pair (a, b) refers to a pair of numbers written in a specific order.

Note:

(a, b) ≠ (b, a)

If (a, b) = (c, d), then a = c and b = d

Coordinate axes

A point's location in a plane is defined by its position in relation to two perpendicular lines known as the coordinate axes.

There are two coordinate axes: X - axis (horizontal axis) and Y - axis (vertical axis).

Origin O (0, 0) is called the point of intersection of the coordinate axes.

In point (a, b), ‘a’ known as abscissa represents the horizontal position and ‘b’ known as ordinate represents the vertical position.

⇒ The axes divide the plane into four quadrants:

Quadrant I:
In this quadrant, both a and b are positive (a > 0 and b > 0).
The ordered pair (a, b) belongs to this quadrant.

Quadrant II:
In this quadrant, a is negative and b is positive (a < 0 and b > 0).
The ordered pair (−a, b) belongs to this quadrant.

Quadrant III:
In this quadrant, both a and b are negative (a < 0 and b < 0).
The ordered pair (−a, −b) belongs to this quadrant.

Quadrant IV:
In this quadrant, a is positive and b is negative (a > 0 and b < 0).
The ordered pair (a, −b) belongs to this quadrant.

Note: To determine the quadrant in which the ordered pair (a, b) belongs, we need to look at the signs of the coordinates a and b.

The coordinates of any point on the x-axis is of the form (a, 0) with y-coordinate zero.

The coordinates of any point on the x-axis is of the form (0, b) with x-coordinate zero.

Points on a Cartesian Plane

A pair of numbers locate points on a plane called the coordinates. The measure of a point's distance from the y-axis is termed the abscissa or x-coordinate. The distance from the x-axis is known as the ordinate or y-coordinate.

Representation of P(x, y) on the cartesian plane.

Distance Between Two Points on the Same Coordinate Axes

The distance between two points that are on the same coordinate axis is given by the difference between their abscissa if they are on the x-axis or by the difference between their ordinates if they are on the y-axis.

Distance Between Two Points Using Pythagoras Theorem

Distance Formula

The distance formula is used to find the distance between two points.
Distance between any two points (x1, y1) and (x2, y2) is given by:

Where d is the distance between the points (x1, y1) and (x2, y2)

Example: What is the distance between the points (2, − 5) and (− 4, 3)?

a) 9 units
b) 10 units
c) 11 units
d) √110 units

Answer: b) 10 units

Explanation: We know that the distance between any two points (x1, y1) and (x2, y2) is given by

Here,

x₁ = 2
y₁ = − 5
x₂ = − 4
y₂ = 3

Thus, the distance between the points (2, − 5) and (− 4, 3) is

d = √[(− 4 − 2)² + (3 − (− 5)²]
= √[(− 6)² + (3 + 5)²]
= √[(− 6)² + (8)²]
= √[36 + 64]
= √100
= 10 units

Section Formula

The section formula is a formula used to find the coordinates of a point that divides a line segment joining two given points in a given ratio.

Internally

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:

Example: What are the coordinates of the point which divides the line segment joining the points (2, –3) and (5, 4) in the ratio 5 : 2 internally?

a) (29/7, 2)
b) (29/7, –2)
c) (–29/7, 2)
d) (–29/7, –2)

Answer: a) (29/7, 2)

Explanation: Here,

x1 = 2
y1 = −3
x2 = 5
y2 = 4
m = 5
n = 2

Let P(x, y) divides the line segment joining the points (2, –3) and (5, 4) in the ratio 5 : 2 internally.

Thus, the coordinates of P are:

Externally

If the point P (x, y) divides the line segment joining A (x1, y1) and B (x2, y2) externally in the ratio m : n, then the coordinates of P are given by the section formula as:

The Sectional Formula is given as:

Example: What are the coordinates of the point which divides the line segment joining the points (1, 4) and (7, 9) in the ratio 3 : 1 externally?

a) (11, 11)
b) (10, 7)
c) (10, 11)
d) (11, 8)

Answer: b) (10, 7)

Explanation: If the point M (x, y) divides the line segment joining P (x1, y1) and Q (x2, y2) externally in the ratio m : n, then the coordinates of M are given by

Here, x1 = 1

y₁ = 4
x₂ = 7
y₂ = 9
m = 3
n = 1

Let P(x, y) divides the line segment joining the points (1, 4) and (7, 9) in the ratio 3 : 1 externally. Thus the coordinates of P are

Thus, P (10, 7) divides the line segment joining the points (1, 4) and (7, 9) in the ratio 3 : 1 externally.

Finding Ratio When The Points Are Given

To find the ratio in which a given point A (x, y) divides the line segment joining P (x1, y1) and Q (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.

If we know the value of x, x1 and x2 then k can be calculated.

Similarly, the value of k can be calculated from the y-coordinate.

Midpoint Formula

The coordinates of the midpoint of a line segment joining A(x1, y1) and B(x2, y2) is given by:

Example: What is the midpoint of a line segment formed by joining (3, 2) and (2, −5)?

a) (2, -3/2)
b) (5/2, 7/2)
c) (5/2, -3/2)
d) (5/2, 3/2)

Answer: c) (5/2, -3/2)

Explanation:

Here,

x1 = 3
y1 = 2
x2 = 2
y2 = −5

Thus, the coordinates of the midpoint of a line segment joining (x1, y1) and (x2, y2) is:

Points of Trisection

Points of trisection refer to points that divide a line segment into three equal parts.

If A and B are two endpoints of a line segment, the points of trisection would divide AB into three equal parts, creating three smaller segments of equal length.

Centroid of a Triangle

Example: What are the coordinates of a centroid of a triangle whose vertices are (4, 8), (− 2, − 9) and (3, −5)?

a) (5/3, -2)
b) (4/3, -2)
c) (5/3, -5/3)
d) (4/3, -5/3)

Answer: a) (5/3, -2)

Explanation: Here,

x1 = 4
x2 = − 2
x3 = 3
y1 = 8
y2 = − 9
y3 = − 5

Thus, the coordinates of centroid (G) of a triangle are:

Area of a Triangle from Coordinates

Example: What is the area of the △ABC if its vertices are A (3, 9), B (−2, 4) and C (−5, −7)?

a) 21 sq units
b) 20 sq units
c) 20.5 sq units
d) 21.5 sq units

Answer: b) 20 sq units

Explanation: Here,

x₁ = 3
x₂ = −2
x₃ = −5
y₁ = 9
y₂ = 4
y₃ = −7

Thus, the area of the triangle is:

Area = ½[x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)]

= ½[3(4 − (−7)) + (−2)(−7 − 9) + (−5)(9 − 4)]
= ½[3(4 + 7) + (− 2)(−16) + (− 5)(5)]
= ½[3(11) + (32) + (− 25)]
= ½[33 + 32 − 25]
= ½[40]
= 20

Thus, the area of the triangle ABC = 20 sq units

Collinearity Condition

If three points A, B and C are collinear and B lies between A and C, then

Where AB, BC and AC are calculated using the distance formula.

→ The area of a triangle formed by three collinear points is zero.