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Complete Math Coordinate
Geometry IGCSE NOTES (Updated
Syllabus Version)
What are coordinates? A number line is useful because it lets you represent a single number. Here are the numbers 5, 2, 0, โ3 and โ4. Now imagine that you take two number lines. You keep one running horizontally and turn the other so that it runs vertically. They cross at 0 on each number line and form a two-dimensional grid. If you want to represent a point on that grid, then you can use a pair of numbers called coordinates. For example, the coordinates (3, 4) represent a point on the grid as shown.
The grid is called a Cartesian plane. It has two intersecting lines: ๏ท an x -axis , which runs horizontally ๏ท a y -axis , which runs vertically The two axes meet at the origin, which is represented by (0, 0) on the grid.
This Cartesian plane shows the positive and negative coordinates in the different regions of the grid. Shapes on a Cartesian plane Which of these sets of coordinates describes a rectangle? a) A (3, 2), B (โ2, 2), C (โ2, โ1), D (3, โ1) b) A (3, 2), B (โ2, 2), C (โ2, 1), D (3, โ1)
c) A (3, 2), B (โ2, โ2), C (โ2, โ1), D (3, โ1) d) A (3, 2), B (โ2, 2), C (โ2, โ1), D (โ3, โ1) Solution: a) A (3, 2), B (โ2, 2), C (โ2, โ1), D (3, โ1) Plot the points A , B , C and D. Only this set of coordinates will give a rectangle.
How can you find the distance between P and Q? To work this out, you need to draw a right-angled triangle using the line PQ as the hypotenuse. Then, find the length of the hypotenuse using Pythagorasโ theorem. For a right-angled triangle with lengths a , b and c as shown, a 2
Find the length of the line segment that joins A(-1, 2) and B(2, 6). Hint: First, find the difference between the &-coordinates and the difference between they-coordinates. How can you use these two numbers to find the distance between the points A and B? You might find it helpful to sketch a coordinate grid, plot the two points and draw a line between them.
Midpoint of a line segment Finding the midpoint of a line segment on a coordinate grid The image below shows points A and B on a grid, with the line segment AB drawn between them. What are the coordinates of the midpoint of AB? The midpoint of AB is the point (3, 2). The image below shows two points C and D on a grid, with the line segment CD drawn between them. Can you work out the coordinates of the midpoint of CD?
The midpoint of CD is the point (1, 1). The midpoint of a diagonal line on a coordinate grid. When finding the midpoint of a vertical or horizontal line, you only have to find the midpoint of two numbers. For diagonal lines, like line segment CD, you need to consider both the x- coordinates and both the y-coordinates. To find the midpoint of line segment CD:
Solution:
What is a gradient? Gradient is another word for 'slope': the greater a line's gradient, the steeper the line is. The image below shows two lines on the same Cartesian plane. The gradient of line A is greater than the gradient of line B because line A is steeper. Gradient formula is:
Which of these is the correct equation for this straight-line graph? a) y = x โ 2
b) y = 2 x โ 1 c) y = 2 x + 1 d) y = x + 2 Answer: Gradient: m = 1 y -intercept: c = โ Substitute these values into y = mx + c : y = 1 x โ 2 or y = x โ 2 Question 1) Find the equation of the line which has gradient 3 and y-intercept -1. Question 2) A straight line passes through the points (1,4) and (-3,-4). Find the equation of the line in the form y=mx+c
- Sketch the graph with equation y=-2x+1 between x=-2 and x=2 by using the gradient and the y-intercept.
Equations of parallel lines Here are two parallel lines on a grid. Parallel lines always remain the same distance apart, and have no intersection point.
