Coordinate Geometry

7.1 Introduction

Coordinate Geometry is a branch of mathematics that combines algebra and geometry to study geometric figures using algebraic methods. It is based on the idea of representing points on a plane using a pair of perpendicular coordinate axes, usually called the x-axis (horizontal) and y-axis (vertical). Each point on the plane is uniquely identified by an ordered pair of numbers (x, y), where x is the x-coordinate (or abscissa) representing the distance from the y-axis measured along the x-axis, and y is the y-coordinate (or ordinate) representing the distance from the x-axis measured along the y-axis. Points lying on the x-axis have coordinates of the form (x, 0), and points on the y-axis have coordinates of the form (0, y). This chapter revisits these concepts and explores important topics such as finding the distance between two points, the midpoint of a line segment, the section formula, and the area of triangles using coordinates. It also connects algebraic equations to geometric representations, such as the fact that a linear equation in two variables represents a straight line on the coordinate plane. The chapter emphasizes the practical applications of coordinate geometry in fields like physics, engineering, navigation, seismology, and art. An introductory activity invites students to plot given points on graph paper and join them in a specified order to form various geometric shapes, helping visualize the coordinate system and the connection between algebraic points and geometric figures.

• Coordinate axes (x and y) are perpendicular and used to locate points on a plane.
• x-coordinate (abscissa) is the distance from the y-axis; y-coordinate (ordinate) is the distance from the x-axis.
• Points on x-axis have coordinates (x, 0); points on y-axis have coordinates (0, y).
• Linear equations in two variables represent straight lines on the coordinate plane.
• Coordinate geometry links algebra and geometry, enabling algebraic study of geometric figures.
• Widely applied in physics, engineering, navigation, seismology, and art.
• 📌 Coordinate axes: Two perpendicular lines (x-axis and y-axis) used to locate points on a plane.
• 📌 Abscissa: The x-coordinate of a point, distance from the y-axis.
• 📌 Ordinate: The y-coordinate of a point, distance from the x-axis.

7.2 Distance Formula

The distance formula is a fundamental tool in coordinate geometry used to find the length of the line segment joining two points in the plane. It is derived from the Pythagoras theorem. Consider two points A and B with coordinates A(x₁, y₁) and B(x₂, y₂). The distance between these points is the length of the segment AB. If the points lie on the same axis, the distance is simply the absolute difference of their coordinates. For example, if both points lie on the x-axis, the distance AB = |x₂ - x₁|; if on the y-axis, AB = |y₂ - y₁|. For points not on the same axis, the distance is found by constructing a right triangle with AB as the hypotenuse. By drawing perpendiculars from points A and B to the x-axis, we form a right triangle where the horizontal leg is |x₂ - x₁| and the vertical leg is |y₂ - y₁|. Applying the Pythagoras theorem, the distance AB is given by: Distance AB = √[(x₂ - x₁)² + (y₂ - y₁)²] This formula always gives a non-negative value since distance cannot be negative. Special cases include the distance of a point from the origin O(0,0), which is √(x² + y²). Several examples illustrate the use of the distance formula: - Checking if three points form a triangle by verifying the triangle inequality. - Determining the type of triangle (e.g., right-angled) using the converse of the Pythagoras theorem. - Verifying if four points form a square by checking equality of sides and diagonals. - Checking collinearity of points by comparing sums of distances. - Finding the locus of points equidistant from two given points, which is the perpendicular bisector of the segment joining them. - Finding a point on an axis equidistant from two given points. This section also includes an activity where students calculate distances between various points and verify geometric properties.

• Distance between two points on the same axis is the absolute difference of their coordinates.
• Distance between any two points A(x₁, y₁) and B(x₂, y₂) is √[(x₂ - x₁)² + (y₂ - y₁)²].
• Distance formula is derived using the Pythagoras theorem applied to the right triangle formed by the points and their projections on axes.
• Distance of a point (x, y) from origin (0,0) is √(x² + y²).
• Used to check if points form a triangle, type of triangle, or other geometric properties.
• Points equidistant from two points lie on the perpendicular bisector of the segment joining them.
• 📌 Distance formula: A formula to calculate the length between two points in the coordinate plane.
• 📌 Pythagoras theorem: In a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides.
• 📌 Collinear points: Points lying on the same straight line.