Class 10 Maths Coordinate Geometry Notes

Class 10 Maths Coordinate Geometry Notes

Class 10 Mathematics — Chapter 7: Coordinate Geometry Notes

1. Coordinate Basics

A point is represented as (x, y).

  • x-coordinate (abscissa): distance from the y-axis.
  • y-coordinate (ordinate): distance from the x-axis.
  • A point on the x-axis has the form (x, 0).
  • A point on the y-axis has the form (0, y).
  • The origin is O(0, 0).

Coordinate geometry connects algebra with geometry and allows geometric properties to be studied using coordinates.

2. Distance Formula

Distance between two points

If

P(x₁, y₁) and Q(x₂, y₂)

are two points, then

PQ = √[(x₂ − x₁)² + (y₂ − y₁)²]

The formula comes directly from the Pythagorean theorem.

Important special case

Distance of P(x, y) from the origin:

OP = √(x² + y²)

Quick memory trick

Distance = √[(difference of x)² + (difference of y)²]

Signs matter while subtracting coordinates, but squaring removes the final sign.

3. Using Distance Formula in Geometry

The distance formula is useful for proving geometric facts.

A. Checking whether three points form a triangle

Find the three side lengths.

For three points to form a triangle:

Sum of any two sides > the third side.

If this condition fails, the points are collinear.

B. Checking for a right triangle

If the side lengths satisfy

a² + b² = c²

then, by the converse of Pythagoras theorem, the triangle is right-angled.

C. Checking whether points are collinear

For points A, B and C lying on the same line:

AB + BC = AC

provided B lies between A and C.

This was used in the chapter to establish collinearity.

4. Proving a Quadrilateral is a Square

One method used in the chapter:

  1. Find all four side lengths.
  2. Show

AB = BC = CD = DA

  1. Find both diagonals and show

AC = BD

A quadrilateral with four equal sides and equal diagonals is a square.

Alternative method

Show that:

  • all four sides are equal, and
  • one angle is 90°.

Then the quadrilateral is a square.

5. Equidistant Points

Suppose P(x, y) is equidistant from A and B.

Then:

PA = PB

It is usually easier to use:

PA² = PB²

because the square roots disappear.

General approach

  1. Write the distance expressions.
  2. Equate their squares.
  3. Expand and simplify.
  4. Obtain the required relation between x and y.

Geometrically, the set of points equidistant from A and B lies on the perpendicular bisector of AB.

6. Finding an Equidistant Point on an Axis

Point on x-axis

Any point on the x-axis is:

P(x, 0)

Point on y-axis

Any point on the y-axis is:

P(0, y)

Then use the condition

PA² = PB²

to find the unknown coordinate.

7. Section Formula

The section formula gives the coordinates of a point that divides a line segment in a specified ratio.

Let

A(x₁, y₁) and B(x₂, y₂)

and let P divide AB internally in the ratio

m₁ : m₂

i.e.

PA : PB = m₁ : m₂

Then

Coordinates of P

P = ( (m₁x₂ + m₂x₁)/(m₁ + m₂),
(m₁y₂ + m₂y₁)/(m₁ + m₂) )

This is the section formula.

Memory trick

The coefficient of a coordinate is attached to the opposite endpoint:

  • x-coordinate → m₁x₂ + m₂x₁
  • y-coordinate → m₁y₂ + m₂y₁

8. Ratio k : 1

If P divides AB internally in the ratio

k : 1

then

P = ( (kx₂ + x₁)/(k + 1),
(ky₂ + y₁)/(k + 1) )

This is simply the section formula with m₁ = k and m₂ = 1.

9. Midpoint Formula ⭐

The midpoint divides a line segment in the ratio 1 : 1.

For

A(x₁, y₁), B(x₂, y₂),

the midpoint is

M = ( (x₁ + x₂)/2, (y₁ + y₂)/2 )

In other words:

Midpoint = average of the two x-coordinates and average of the two y-coordinates.

10. Trisection of a Line Segment

To divide AB into three equal parts, there are two points.

If

A → P → Q → B,

then:

  • P divides AB in ratio 1 : 2
  • Q divides AB in ratio 2 : 1

Use the section formula for each point.

11. Finding the Ratio in Which a Point Divides a Line

If the coordinates of A, B and an internal point P are known:

  1. Assume the ratio is m₁ : m₂ (or k : 1).
  2. Apply the section formula.
  3. Equate the x-coordinates.
  4. Solve for the ratio.
  5. Verify using the y-coordinate.

The chapter demonstrates this method for finding an unknown division ratio.

12. Point Where a Line Segment Meets an Axis

This is a common application of the section formula.

If the point lies on the y-axis:

Its x-coordinate must be 0.

If the point lies on the x-axis:

Its y-coordinate must be 0.

So:

  1. Assume the division ratio.
  2. Apply the section formula.
  3. Put the appropriate coordinate equal to zero.
  4. Solve for the ratio.
  5. Substitute back to find the point.

The chapter uses this method to find where a line segment intersects the y-axis.

13. Parallelogram Problems

A key property:

The diagonals of a parallelogram bisect each other.

Therefore, the midpoint of one diagonal is equal to the midpoint of the other.

For example, if ABCD is a parallelogram:

Midpoint of AC = Midpoint of BD

This can be used to find an unknown coordinate.

14. External Division

The section formula given in this chapter is for internal division, where P lies between A and B.

If P lies outside the line segment AB but still on the same straight line, it divides AB externally.

The chapter notes that the external section formula is studied in higher classes.

Formula Sheet ⭐

Distance Formula

PQ = √[(x₂ − x₁)² + (y₂ − y₁)²]

Distance from Origin

OP = √(x² + y²)

Section Formula

For PA : PB = m₁ : m₂:

P = ( (m₁x₂ + m₂x₁)/(m₁+m₂),
(m₁y₂ + m₂y₁)/(m₁+m₂) )

Midpoint Formula

M = ( (x₁+x₂)/2, (y₁+y₂)/2 )

These are the main formulas summarised by the chapter itself.

Exam Strategy

When you see a question…

“Find distance”
→ Use Distance Formula.

“Show points are collinear”
→ Compare distances, usually using AB + BC = AC.

“Show triangle is right-angled”
→ Check a² + b² = c².

“Show quadrilateral is a square”
→ Compare four sides and diagonals, or use equal sides + one right angle.

“Point divides a line in ratio…”
→ Use Section Formula.

“Find midpoint”
→ Use Midpoint Formula.

“Point lies on x-axis”
→ Its y-coordinate = 0.

“Point lies on y-axis”
→ Its x-coordinate = 0.

“Parallelogram”
→ Use the fact that diagonals bisect each other.

“Equidistant”
→ Set the squares of distances equal.

Most Important Things to Memorise

  1. Distance Formula
  2. Section Formula
  3. Midpoint Formula
  4. x-axis → y = 0
  5. y-axis → x = 0
  6. Equidistant → PA = PB
  7. Collinear → appropriate distances satisfy the line condition
  8. Right triangle → Pythagorean theorem
  9. Parallelogram → diagonals bisect each other
  10. Internal section formula only; external division is not developed in this chapter.