Coordinate Geometry | Class 10 Mathematics Notes

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.

📊 Diagram: Fig. 7.1 shows the graphical representation of two towns A and B with coordinates; Fig. 7.2 illustrates points on axes and their distances; Fig. 7.3 and Fig. 7.4 demonstrate distance calculation using perpendiculars and Pythagoras theorem; Fig. 7.5 shows the general case of distance between two points.

🧪 Activity: Calculate distances between given pairs of points, check for triangle formation, verify types of triangles, and find points equidistant from given points using the distance formula.

🔗 Connection: This section provides the foundational formula for measuring distances between points, which is essential for understanding the Section Formula in the next section that deals with dividing line segments.