CBSE Class 10 Maths Notes: Coordinate Geometry

Cartesian Coordinate Plane & Plotting Points

Let’s dive into the world of coordinate geometry! This chapter introduces us to representing points and shapes on a plane using numbers. We’ll start with the foundation: the Cartesian coordinate plane.

Definitions:

• The Cartesian Plane is a two-dimensional plane formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical).
• The point where the x-axis and y-axis intersect is called the origin, denoted by the coordinates (0, 0).
• Each point on the plane is uniquely identified by an ordered pair of numbers (x, y), called coordinates.
• x-coordinate (or abscissa): Represents the horizontal distance from the y-axis.
• y-coordinate (or ordinate): Represents the vertical distance from the x-axis.

Plotting Points: To plot a point (x, y), we move:

• x units along the x-axis (right if x is positive, left if x is negative).
• y units along the y-axis (up if y is positive, down if y is negative).

Distance Formula

A crucial concept is calculating the distance between two points in the Cartesian plane. The distance formula allows us to do just that!

Core Principles: The distance formula is derived using the Pythagorean theorem. Imagine forming a right-angled triangle with the line segment connecting two points as the hypotenuse. The legs of the triangle are parallel to the x and y axes.

Formula: The distance, *d*, between two points $A(x_1, y_1)$ and $B(x_2, y_2)$ is given by: $$d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}$$

Applications:

• Calculating the length of a line segment.
• Determining if three points are collinear (lie on the same line). If the sum of the distances between two pairs of points equals the distance between the remaining pair, the points are collinear.
• Proving geometric properties (e.g., sides of a triangle, diagonals of a square).

Section Formula (Internal Division)

Now, let’s explore how to find the coordinates of a point that divides a line segment internally in a specific ratio.

Formula: If a point $P(x, y)$ divides the line segment joining the points $A(x_1, y_1)$ and $B(x_2, y_2)$ internally in the ratio $m:n$, then the coordinates of P are: $$x = \frac{m x_2 + n x_1}{m + n}, \quad y = \frac{m y_2 + n y_1}{m + n}$$

Explanation:

• The ratio $m:n$ represents the ratio in which the point P divides the line segment AB.
• The formula provides a direct way to calculate the x and y coordinates of the dividing point.

Examples:

• Finding the midpoint of a line segment (m:n = 1:1). In this case, the coordinates become: $x = \frac{x_1 + x_2}{2}, y = \frac{y_1 + y_2}{2}$
• Finding a point that divides a line segment in a ratio like 2:3.

Applications in Geometry

Let’s see how we can use the distance formula and section formula to solve problems in geometry.

Problems:

• Verifying the type of triangle: Show that the points A, B, and C form an equilateral, isosceles, or scalene triangle by calculating the lengths of the sides using the distance formula.
• Proving properties of quadrilaterals: Show that the points A, B, C, and D form a square, rectangle, rhombus, or parallelogram by calculating the lengths of sides, diagonals, and checking for perpendicularity or equality.
• Finding the centroid of a triangle: The centroid is the point of intersection of the medians (lines from a vertex to the midpoint of the opposite side). The centroid’s coordinates are the averages of the vertices’ coordinates. $Centroid = \left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}\right)$
• Other problems include: finding the ratio in which a point divides a line segment, proving collinearity, and finding the coordinates of a point given specific geometric conditions.

Tips for Solving Problems:

• Draw a diagram: Visualize the problem by drawing a diagram to represent the given information.
• Identify key formulas: Determine which formulas (distance or section) are applicable.
• Organize your work: Clearly state the given information and what you need to find. Show your calculations step-by-step.
• Check your answer: Does your answer make sense in the context of the problem?