## Ratios and Geometry

Ratios are used a lot in geometry, especially when comparing shapes of different sizes and also lengths of the sides of shapes.  In a lot of situations you can find equivalent ratios in a diagram, which can help you find how long a side of a shape is.

### Proportional or similar shapes

Check out these two rectangles:

What is the ratio between the width and the height of the smaller rectangle?  Well, it’s 2 cm wide and 4 cm high, so the width:height ratio is 2:4, which simplifies to 1:2.  What about for the larger rectangle?  It’s 4 cm wide and 8 cm high, so the width:height ratio is 4:8 which simplifies to 1:2.

Notice how both rectangles have the same width to height ratio.  Mathematicians would describe these rectangles as being in proportion to one another.  Or you could just say that the two rectangles are proportional.  Another word which means the same as proportional is similar.  So these two rectangles are also similar.

Similar shapes don’t have to be lined up nicely, which sometimes makes your job identifying them a bit harder.  For instance, these two shapes are similar:

Because the small shape has been rotated quite a bit compared to the larger one, it’s hard to compare the two in your mind.  One way to compare them in your mind is to mentally rotate the smaller shape until it’s at the same angle as the larger one, and then mentally place it inside the larger one like this:

Getting good at doing this rotation in your mind is important, because it can help you avoid getting tricked by situations like this:

You might think that these two shapes are similar, since they both look like block letter ‘L’s.  However, just in case, let’s rotate the smaller shape and try and align it inside the larger one:

No matter what way we rotate the smaller shape, it never looks like a smaller version of the large one.  The closest it gets is at the 2nd attempt, but even then the long part of the small ‘L’ is in the short part of the large ‘L’, and vice versa.

Similar or proportional shapes have the same shape, but are different sizes.  There is however a word for shapes that are exactly the same:

### Congruent shapes

If you have two identical shapes:

they are said to be congruent.  Congruent shapes are the exact same size and the exact same shape.  Because they have the exact same size and shape, this means that congruent shapes must have equal angles and equal side lengths as well.

### How to tell if two shapes are congruent

There are a few tests you can use to decide whether two shapes are congruent.

Can you move or rotate one shape so that it covers the other shape exactly?

Is one shape the mirror image of the other?

These are congruent shapes, since they are mirror images of each other.  Remember that to be a mirror image, they must be the same size.

Are all corresponding sides lengths and angles the same?

These two triangles are congruent, since all corresponding angles and side lengths are the same. Corresponding means you have to compare the same bits in each triangle, for instance the shortest side in one triangle corresponds to the shortest side in the other triangle.

You can also use some general knowledge and reasoning to help you work out whether two shapes are congruent.  For instance, with triangle shapes, there are 4 things you can look for to tell whether two triangles are congruent:

### Congruent triangles

Two sides are equal in length, and the angles included by these sides (in between them) are equal.

This is known as SAS – (S)ide (A)ngle (S)ide.

A pair of angles on one triangle is equal to a pair of angles on the other triangle, and the side between these angles in one triangle is equal in length to the corresponding side in the other triangle.

This is known as ASA - (A)ngle (S)ide (A)ngle.

Three sides on one triangle equal in length to corresponding three sides on the other triangle.

This is known as SSS – (S)ide (S)ide(S)ide.

For right-angled triangles you don’t need as much information as to whether they are congruent or not.  All you need is two corresponding sides to be the same length: