Auckland Sports College
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"Moving Forward Together"
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Co-ordinate Geometry
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The topics in Co-ordinate Geometry are
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As well as these textbooks there is a useful booklet produced by mathscentre.co.nz. It can be downloaded here
The Distance Between Two Points
This is just Pythagorus. In a right angled triangle the square of the longest side equals the sum of the squares of the two shorter sides.
(x2 - x1) is just the distance AC, or the change in the x values
(y2 - y1) is just the distance BC, or the change in the y values
The midpoint of a line segment
The midpoint of a line segment is just the average of the x values and the average of the y valuesFind the midpoint of (1,7) and (5, 9)
The average of the x values is (1+5)/2 = 3
The average of the y values is (7+9)/2 = 8
The midpoint is (3,8)
The gradient (slope) of a line
The slope is rise/run or the change in the y values divided by the change in the x valuesFind the slope of the line joining (1,7) and (5, 9)
The rise or change in y is 9 - 7 = 2
The run or change in x is 5 - 1 = 4
The slope (or gradient), m, is 2/4 = 0.5
The Equation of a Line
We use the formula for a straight line y = mx + c
where 'm' is the slope or gradient (rise/run), 1.2 in this example
and where 'c' is the y-intercept (the value of 'y' when x = 0
so, y = 1.2x + c and (3, 11) is on the line
11 = 1.2*3 + c
11 = 3.6 + c
c = 7.4
The equation of the line is y = 1.2x + 7.4
An alternative formula is
Parallel Lines
Perpendicular Lines
The Intersection of Two Lines
For this standard we use substitution to answer these questionsFind the intersection of y = 2x + 7 and y = -x +4
As both equations are true at the intersection we can substitute '-x + 4' for 'y' in the first equation
2x + 7 = -x + 4
3x + 7 = 4 (adding 'x' to both sides)
3x = -3 (subtracting 7 fron both sides
x = -1 (dividing both sides by 3) and as y = 2x + 7, y = -2 + 7, y = 5
The lines intersect at (-1, 5)