1. Co-Ordinate Geometry
Maths Studies

2. Distance Learning Objectives
To find the distance of a line segment using two co-ordinates.
To find coordinates when given 1 coordinate and the distance value.
Determine the shape of a shape e.g. triangle using the distance formula.

3. Distance
Example 1: Use the Distance Formula to find the distance between the points with coordinates (−3, 4) and (5, 2).

4. Distance √(-1- -2)²+(q – 4)² = √10
Example
Find q given that P(-2,4) and Q(-1,q) are √10 units apart.
√(-1- -2)²+(q – 4)² = √10
√(-1)² + (q – 4)² = √10 Square both sides to remove √
(-1)² + (q – 4)² = 10
1 + (q-4)² = 10
(q-4)² = 10 – 1
(q – 4)² = 9
(q – 4)² = √9 To remove the √ find the √ of the other side
q – 4 = 3
q = 7

5. Midpoint Learning Objective
To find the midpoint of a line segment using two co-ordinates.
To find coordinates when given 1 coordinate and the midpoint.

6. Midpoint
Find the midpoint R of the line segment with coordinates (-9,-1) and (-3,7)

7. Midpoint

8. Gradient
The gradient of a line is a measurement of the steepness and direction of a nonvertical line.

9. Gradient Learning Objectives
To find the gradient of a line given two coordinates.
To find coordinates when given 1 coordinate and the gradient.
Understand the relationship of parallel and perpendicular lines.
Find the gradient of parallel and perpendicular lines

10. Gradient Formula Example M = Y2 – Y1 X2 – X1 A(1,5) B(2,8)
2 -1

11. Drawing a Line
Draw a line that passes through the point A(1,4) with gradient -1 (see pg. 91 eg 2)

12. Gradient Perpendicular Lines
Perpendicular lines have opposite gradients.
Eg Gradient of AB = 2/3
Gradient of CD = -3/2

13. Gradient Parallel Lines Parallel lines have the SAME gradient
e.g. Gradient of MN = 3/9
Gradient of OP = 1/3

14. Lines
To prove that two lines are perpendicular, simply multiply their gradients. If the answer is -1, then the lines are perpendicular.
Eg. The gradient of line AB is 2/3
The gradient of line CD is -3/2
2 x - 3

15. Equation of a line
An equation of a line can come in different forms depending on their gradient, if they are horizontal or vertical and if they intercept the y-axis.

16. Equation of a line Example 1 – Gradient Intercept Form y = mx + c
Where m = gradient
and c is the y-intercept
y = mx + c
(0,c)
intercept
See Example 5 P. 95

17. Equation of a Line Example 2 – General Form ax + by + c = 0
To find the equation of a line in this form you need a gradient and one point. However, to find a gradient you need two points.

18. Equation of a line
E.g. Find the equation of a line that joins the points
A(-3,5) B(1,2)
The gradient is m = 2 – 5 = - 3
1-(-3)
Using this formula y – y1=m(x – x1)
y - -3 = -3/4(x – 5)
4(y + 3) = -3(x – 5)
4y + 12 = -3x + 15
3x + 4y – 3 = 0

19. Equation of a Line y - -3 = -3/4(x – 5) 4(y + 3) = -3(x – 5)
Ans = 3x + 4y – 3 = 0