Help with Understanding Geometry

Geometry is easier than you think. Plane geometry is based on concepts that on their own are extremely simple; if you can draw a line, you can do geometry. So, whether you need some help with homework or you’d just like to understand geometry a bit more, here’s a guide to understanding it.

Firstly, lets go right back to basics. Draw two lines on a sheet of paper; one going vertically down the left hand side and the other horizontally along the bottom. With a ruler, put a small mark every centimetre on both lines. Already, you have in front of you the basis of all geometry. With geometry, using nothing more than these two lines you can describe any location on the page.

For example, we can describe the exact position of a dot on the page (technically now a two-dimensional ‘plane’) with just two numbers. These numbers are called co-ordinates (“co” means “together”, since we always need both numbers together), and are written in brackets like this: (4, 7).

This simply tells us where our dot is in relation to our lines. To find our dot, you could put a pen at the place where the two lines cross, called the ‘origin’, and follow the numbers like instructions. (4, 7) therefore means we move the pen 4 units to the right, then 7 units towards the top of the page. Simple! We just located a dot, technically called a ‘point’, using just two numbers. In fact, we could even use negative numbers to describe a point that is to the left of the vertical line or below the horizontal line. For example, (5, -7) means ‘five units right and seven units down’.

The two lines we drew are called ‘axes’ – a horizontal axis and a vertical axis. To make it quicker to write down, we can give them a single letter instead. The standard convention is to call the horizontal axis ‘x’ and the vertical one ‘y’. If you’re wondering where ‘z’ has got to, we’ll mention this later.

Now we have a system for describing locations in the form of points we can describe more complex objects. A point is known as zero-dimensional because it has no size; it is just the name given to a place precisely at the co-ordinates we want. If we draw two points and join them up we now have a one-dimensional line segment. It’s only dimension is length (of course, the pencil line has width so we can see it but in theory it has no width).

We can also define lines by an equation that tells us what the co-ordinates will be. For example, we know that (4, 7) means x=4 and y=7, meaning ‘four along and seven up’. Now I can describe a line as y=4. This is a simple way of saying the line contains the points (0, 4), (1, 4), (2, 4) and so on. Whatever x is, y will always be 4 and if we join these points up we get a straight line that goes through the y-axis right on the number four. We can do the same for the x-axis: x=6 will be a vertical line that goes through the x-axis on the sixth mark.

The same concept can be used in geometry to describe any line, it just depends how adventurous you want to get with the equation! For example, y=2x simply means whatever number you go across, you go twice as many up. The points (1, 2), (2, 4) and (3, 6) would all be on this line. There is no limit to how complex a line can be, for example, y=17x + 5 simply means whatever x value you choose, the corresponding y value is 17 times larger and has five added.

Lines are infinitely long because we can choose any value for x. This means that any two lines on the same plane will either be parallel (equal distance from each other all the way along) or will eventually meet each other at some point.

If the equation for a line has numbers being squared (x squared will be called x^2 due to Helium’s typesetting limitations) then the line will be a curve. If you try and draw y=x^2 you might not think it looks that way if you only put a dot on (1, 1), (2, 4), (3, 9) and so on. If, however, we take half units into account, for example by putting a dot on (1.5, 2.25), (2.5, 6.25) and so on, it looks a bit more curvy. If we put a dot on every 10th of a unit it would look even more like a curve. If it was possible to put an infinite number of dots on the page the line would be a perfect curve. In practice, though, it’s fine to put in only a few dots and sketch the curve in yourself.

In case you’re wondering, a circle can be drawn by plotting y^2 + x^2 = the radius of the circle squared. Try plotting y^2 + x^2 = 25 and you’ll get a circle with a radius of five units.

There’s no limit to how far we can extend geometry – we can add a third dimension to describe points above or below the page, which would look like (2, 5, 9), and the third axis would be called ‘z’. Or, we could make even more complex line equations like x cubed, x divided by a number, or even a number divided by x. But, as we have just seen, geometry is built on principles that are really quite simple!