Lesson 1.1 Distance, midpoint, gradient

This is a free lesson. We trust you enjoy it!

Introduction

Note: this lesson is longer than most, so don't be afraid that all of them are going to take as long as this!

Please watch this video, which gives you an introduction to the chapter.

Distance between two points in a plane

The first section of the chapter deals with finding the distance between two points in a plane. The following videos explain the idea and show how it is applied. (It's not necessary for you to be able to derive the formula, but you should know it and how to use it.)

For your notebook:

Distance Formula

The distance between two points, A and B, on a coordinate plane is given by the formula:

$$ distance = \sqrt{ (x_B - x_A)^2 + (y_B - y_A)^2} $$

The midpoint of a line segment

We can also use the algebra of coordinates to find the midpoint of a line segment. The following video explains how. (Once again, you don't need to know how to derive the formula.)

For your notebook:

The midpoint formula

The coordinates of the midpoint of a line segment joining points  \(A  (x_A,y_A) \) and  \(B  (x_B,y_B) \) are given by:

 $$(x,y) = \left(  { 1 \over 2 } (x_A + x_B),  { 1 \over 2 } (y_A + y_B) \right) $$

The gradient of a line

Gradient is a common concept in everyday life. For example, a hill may be said to have a steep gradient or a gentle gradient. This idea is also used in mathematics, where we define the gradient of a line precisely, as explained in this video.

For your notebook:

Gradient of a line

The gradient of a line is defined by:

 $$gradient =  { \Delta y \over \Delta x  } =  { y_B - y_A \over x_B - x_A }  $$

where  \( A (x_A, y_A) \) and  \( B (x_B , y_B) \) are any two points on the line.

Combined examples

The following two videos discuss two examples (1.3.1 and 1.3.2) from the text book. Notice how they combine the concepts of distance, gradient, and midpoint, and how — especially in the second example — they apply the concepts to a more complex problem rather than just calculating the quantities.

Weight lifting

Complete the following exercises from your text book:

Exercise no.1A

  • 1 b, d, f, h, j
  • 2, 3, 4
  • 5 a, c, e, g
  • 6, 7
  • 10 a, c, e, g
  • 11, 12, 13, 16, 18, 20, 22

You can download the answer key from here, or from your download area on the My Account menu. Do so, and mark your work as explained in the Preliminaries.