🗺️ Class 9 · Geometry

Coordinate Geometry for Class 9: notes, worked examples and practice

Coordinates give every point on a plane an address (x, y). With the distance and midpoint formulas you can solve geometry problems using only numbers.

Updated 21 October 2026. Every worked example and answer on this page is checked by automated tests.

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Notes

Coordinate Geometry: what you need to know

Chapter names and order differ by board and textbook edition, so follow your own textbook for the order.

The Cartesian plane

Two number lines, the x-axis (horizontal) and the y-axis (vertical), cross at the origin (0, 0). A point is written as (x, y): first how far along, then how far up or down.

Quadrants

The axes split the plane into four quadrants. I: (+, +). II: (−, +). III: (−, −). IV: (+, −). Points on an axis lie in no quadrant.

Distance formula

The distance between (x₁, y₁) and (x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²). It is the Pythagoras theorem on a right triangle drawn between the points.

Midpoint formula

The midpoint of the line joining (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2).

Using the formulas

Use distances to check whether a triangle is isosceles, equilateral or right-angled. Use midpoints to find centres of line segments.

Formulas and rules to remember

Distance = √((x₂ − x₁)² + (y₂ − y₁)²)
Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Signs: I (+,+), II (−,+), III (−,−), IV (+,−)

Solved examples

Example 1

Find the distance between (1, 2) and (4, 6).

x₂ − x₁ = 3 and y₂ − y₁ = 4.
Distance = √(3² + 4²) = √25 = 5.

Answer: 5 units

Example 2

Find the midpoint of the line joining (2, −4) and (8, 6).

x: (2 + 8)/2 = 5.
y: (−4 + 6)/2 = 1.

Answer: (5, 1)

Example 3

In which quadrant does the point (−3, 5) lie?

x is negative and y is positive.
That is the second quadrant.

Answer: Quadrant II

Example 4

Show that the points (0, 0), (3, 4) and (6, 0) form an isosceles triangle.

From (0, 0) to (3, 4): √(9 + 16) = 5.
From (3, 4) to (6, 0): √(9 + 16) = 5.
From (0, 0) to (6, 0): 6.
Two sides are equal, so it is isosceles.

Answer: Two sides are 5, so it is isosceles

Avoid these

Common mistakes

Swapping x and y

(3, 5) and (5, 3) are different points. x comes first.

Sign errors in the differences

Subtract in the same order for x and y, and square so signs do not matter, but be careful before squaring.

Forgetting the square root

After adding the squares you have the distance squared.

Confusing midpoint with distance

The midpoint is a point. The distance is a length.

10 practice questions with answers

Try each question on paper first, then tap it to check your answer. The questions are generated by code and each answer is checked.

Find the distance between the points (−1, 1) and (2, 5). (in units)
Answer: 5
Find the distance between the points (3, −3) and (11, 12). (in units)
Answer: 17
Find the distance between the points (6, 6) and (12, −2). (in units)
Answer: 10
Find the midpoint of the line joining (0, 3) and (0, 13).
Answer: (0, 8)
Find the distance between the points (1, 5) and (−4, 17). (in units)
Answer: 13
Find the midpoint of the line joining (−5, 4) and (−3, 2).
Answer: (−4, 3)
Find the midpoint of the line joining (3, 6) and (5, 8).
Answer: (4, 7)
Find the distance between the points (1, −6) and (6, 6). (in units)
Answer: 13
Find the slope of the line through (−1, −4) and (0, −4):
Answer: 0
Find the slope of the line through (1, 5) and (3, 7):
Answer: 1

More practice

Printable worksheets (PDF with answers)

Slope between two points (Grade 8)

Games and lessons

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FAQ

Coordinate Geometry Class 9: questions students ask

What is the distance formula?

√((x₂ − x₁)² + (y₂ − y₁)²). It comes from the Pythagoras theorem.

What is the midpoint formula?

Average the x-coordinates and the y-coordinates: ((x₁ + x₂)/2, (y₁ + y₂)/2).

Which quadrant is (−2, −5) in?

The third quadrant, because both coordinates are negative.

What are the coordinates of the origin?

The origin is the point where the x-axis and the y-axis meet. Its coordinates are (0, 0).

Which board is this coordinate geometry page for?

It covers the ideas taught in CBSE, ICSE and state-board maths. Chapter names and order differ by board and textbook edition, so follow your own textbook for the exact order.

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