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Chapter 7: Coordinate Geometry

    Introduction

    Distance Between Two Points

    Distance Between Two Points on a Line Parallel To The Coordinate Axis

    Distance Between Any Two Points In the X-Y Plane

    Section Formula

    TriSectional Points of a Line

    Centroid Triangle

    Area of the Triangle

    Straight Lines

    Summary

    Enhanced Curriculum Support

    Easy Level Worksheet

    Moderate Level Worksheet

    Hard Level Worksheet

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Chapter 7: Coordinate Geometry > Distance Between Two Points on a Line Parallel To The Coordinate Axis

Distance Between Two Points on a Line Parallel To The Coordinate Axis

Let's begin by examining the simpler cases where points lie on lines parallel to the coordinate axes, then extend our understanding to find distances between any two points in the plane.

Case 1: Points on a Horizontal Line

Consider two points A(x1, y1) and B(x2, y1). Since both points have the same -coordinate, they lie on a line parallel to the -axis.

To find the distance between these points:

  • Draw perpendiculars AP and BQ to the x-axis

  • Observe that figure APQB forms a rectangle

  • Therefore, AB =

The distance PQ equals the absolute value of the difference between the x-coordinates:

Distance = |x2 - x1|

Note: We use absolute value because distance is always positive, regardless of which point is further to the right.

Case 2: Points on a Vertical Line

Similarly, when two points A(x1, y1) and B(x1, y2) have the same -coordinate, they lie on a line parallel to the -axis.

The distance between these points is: Distance = |y2 - y1|

This represents the absolute value of the difference between the y-coordinates.

Example 1

Problem 1: What is the distance between A(4, 0) and B(8, 0)?

Solution: Since both points have y-coordinate equal to , they lie on the -axis (a line).

Distance = |x2 - x1| = | - | = units

Example 2

Problem: Find the distance between A(8, 3) and B(-4, 3).

Solution: Both points have y-coordinate equal to , so they lie on a line i.e. -axis.

Distance = |x2 - x1| = | - | = || = units

Note: Here x1 and x2 are in different quadrants, but the absolute value ensures our answer is positive.

General Case: Distance Between Any Two Points

When points do not lie on lines parallel to the coordinate axes, we use the Pythagorean theorem.

Developing the Distance Formula

Let's find the distance between A(4, 0) and B(0, 3):

(1) Set up the coordinate system: Place points A and B on the coordinate plane with origin O.

(2) Form a right triangle: The triangle △AOB is a right-angled triangle because:

OA = units (along x-axis)

OB = units (along y-axis)

Angle AOB = °

(3) Apply the Pythagorean theorem:

AB2 = OA2 + OB2 = + = + =

AB = 25 = units

The General Distance Formula

For any two points A(x1, y1) and B(x2, y2), the distance d between them is:

d = x2−x12+y2−y12

This formula works for all cases:

When points are on horizontal lines: (y2 - y1) =

When points are on vertical lines: (x2 - x1) =

When points are anywhere in the plane

Practice Problems

Find the distance between the following pairs of points:

(1) Distance between (3, 8) and (6, 8) = units
(2) Distance between (-4, -3) and (-8, -3) = units
(3) Distance between (3, 4) and (3, 8) = units
(4) Distance between (-5, -8) and (-5, -12) = units
(5) Distance between A(1, 2) and B(4, 6) = units
(6) Distance between P(-2, 3) and Q(1, -1) = units
(7) Distance between M(0, 0) and N(5, 12) = units