📘 Class 10 Maths Chapter 18: Coordinate Geometry – Notes & Blog Post

 

🗺️ Introduction

Coordinate Geometry is the study of geometry using the coordinate plane. This chapter helps us understand the position of points, distance between points, and area of triangles formed on the Cartesian plane.

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🧭 1. Coordinate System

• Origin (O): The point (0, 0) where the x-axis and y-axis intersect.

• X-axis: Horizontal line.

• Y-axis: Vertical line.

• Coordinates of a point (x, y): Represented as:

  • x → horizontal distance

  • y → vertical distance

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📏 2. Distance Formula

Used to find the distance between two points $A(x_1, y_1)$ and $B(x_2, y_2)$:

$$\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

✅ Example:

Find distance between A(3, 4) and B(7, 1):

$$= \sqrt{(7 - 3)^2 + (1 - 4)^2} = \sqrt{16 + 9} = \boxed{5}$$

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📍 3. Section Formula

Divides a line segment joining two points $A(x_1, y_1)$ and $B(x_2, y_2)$ in the ratio $m:n$:

$$\left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n} \right)$$

🔸 Midpoint Formula:

If $m = n = 1$, then it becomes:

$$\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$$

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🧮 4. Area of a Triangle

To find the area of a triangle with vertices $A(x_1, y_1)$, $B(x_2, y_2)$, $C(x_3, y_3)$:

$$\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$$

✅ Example:

A(1, 2), B(4, 6), C(5, 2)

$$= \frac{1}{2} |1(6 - 2) + 4(2 - 2) + 5(2 - 6)| = \frac{1}{2} |4 - 0 - 20| = \boxed{8 \text{ units}^2}$$

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📌 Key Concepts Table

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🔔 Points to Remember

• All points are represented in the form (x, y).

• Use Distance formula for length between 2 points.

• Use Area formula only for triangles with known coordinates.

• In coordinate geometry, signs matter – be careful!

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