Class 9 Maths Exercise 7.2: Coordinate Geometry

Class 9 Mathematics Exercise 7.2 – Coordinate Geometry (Complete Explanation with Examples)

Coordinate Geometry is one of the most important topics in Class 9 Mathematics because it connects algebra with geometry in a very practical way. In Exercise 7.2, students learn how to calculate distances between points, check collinearity, and identify geometric shapes using coordinates. These concepts are frequently tested in board exams, so understanding them properly is essential.

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Understanding the Distance Formula

The distance formula is the foundation of this exercise. It allows us to find the distance between two points on a coordinate plane. If we have two points:

(x₁, y₁) and (x₂, y₂)

The distance between them is calculated using:

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

This formula comes directly from the Pythagorean theorem. Many students try to memorize it, but from teaching experience, it becomes much easier when you understand that it is simply finding the hypotenuse of a right triangle formed between two points.

Example 1

Find the distance between A(-2, 1) and B(1, 5).

d = √[(1 − (-2))² + (5 − 1)²]
d = √[(3)² + (4)²]
d = √(9 + 16)
d = √25 = 5

This type of question is very common in exams, and students often lose marks due to sign mistakes. Always be careful when subtracting negative numbers.

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Collinear Points

Three or more points are called collinear if they lie on the same straight line. To verify this, we use the distance formula.

If points A, B, and C are collinear, then:

AB + BC = AC

Example

Check whether A(0,1), B(2,3), and C(3,4) are collinear.

AB = √[(2−0)² + (3−1)²] = √(4 + 4) = 2√2
BC = √[(3−2)² + (4−3)²] = √(1 + 1) = √2
AC = √[(3−0)² + (4−1)²] = √(9 + 9) = 3√2

Since AB + BC = 2√2 + √2 = 3√2 = AC, the points are collinear.

In exams, students often forget to compare the final values. Always write the conclusion clearly to secure full marks.

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Identifying Geometric Shapes Using Coordinates

One of the most interesting parts of Exercise 7.2 is identifying shapes such as squares, rectangles, and parallelograms using coordinates. Instead of drawing diagrams, we rely on distances.

1. Square

A square has all sides equal.

Example: If AB = BC = CD = DA, then the shape is a square.

In one typical problem, all four sides come out to √29. Since all sides are equal, the figure is a square.

2. Rectangle

A rectangle has opposite sides equal.

If AB = DC and AD = BC, then the figure is a rectangle.

For example, if AB = 6 and DC = 6, and AD = 4 and BC = 4, then the shape is a rectangle.

3. Parallelogram

A parallelogram also has opposite sides equal.

If AB = DC and AD = BC, then the shape is a parallelogram.

In exams, students sometimes confuse rectangles and parallelograms. The key difference is that rectangles have right angles, while parallelograms do not necessarily have them.

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Common Mistakes Students Make

• Forgetting to square negative values properly
• Skipping steps in calculations
• Not writing the final conclusion in collinearity questions
• Confusing properties of square, rectangle, and parallelogram
• Making arithmetic mistakes under square roots

From teaching experience, most mistakes are not conceptual but careless. Writing each step clearly helps avoid these errors.

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Exam Tips for Exercise 7.2

• Always write the distance formula before solving
• Show complete working to gain method marks
• Keep calculations neat and organized
• Double-check signs when subtracting coordinates
• Write a proper conclusion in geometry questions

Examiners award marks for method as well as the final answer, so presentation plays an important role.

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Conclusion

Exercise 7.2 builds a strong foundation in coordinate geometry by teaching distance calculation, collinearity, and shape identification. These concepts are not only important for Class 9 exams but also for future topics like straight lines and graphs.

Once you understand the logic behind each formula and practice regularly, this chapter becomes one of the easiest scoring areas in Mathematics.