Similarity

For rigid transformations, the image is always to the original – but this is true for dilations. Instead, we say that two shapes are similar. They have the same overall shape, but not necessarily the same size.

The symbol for similarity is ∼ (similar to the symbol for congruence, which was ≅). In this example, we would write A∼A’.

COMING SOON – Illustration

Perspective Drawings

You might have noticed that these dilations with the connecting rays almost look like perspective drawings. The center of dilation is called the vanishing point, because it looks like this is where everything is “vanishing in the distance”.

Find the vanishing point in the figure below:

COMING SOON – Interactive

Now can you draw another house that matches the existing ones?

Similar Polygons

Similarity can tell us a lot about shapes. For example, circles, squares and equilateral triangles are similar. They might have different sizes, but always the same general shape.

The two quadrilaterals on the right are similar. Our first important observation is that in similar polygons, all the matching pairs of angles are congruent. This means that

∡ABC ≅ ∡A'B'C' ∡BCD ≅ ∡B'C'D' ∡CDE ≅ ∡C'D'E' ∡DEA ≅ ∡D'E'A'

The second important fact is that in similar polygons, all sides are scaled proportionally by the scale factor of the corresponding dilation. If the scale factor is ${k}, then

AB× ${k} =A’B’BC× ${k} =B’C’ CD× ${k} =C’D’DE× ${k} =D’E’

We can instead rearrange these equations and eliminate the scale factor entirely:

ABA’B’=BCB’C’=ABA’B’=ABA’B’

We can use this to solve real life problems that involve similar polygons – for example finding the length of missing sides, if we know some of the other sides. In the following section you will see a few examples.

Similar Triangles

The concept of similarity is particularly powerful with triangles. We already know that the corresponding internal angles in similar polygons are equal.

For triangles, the opposite is also true: this means that if you have two triangles with the same three angle sizes, then the triangles must be similar.

And it gets even better! We know that the internal angles in a triangle always add up to °. This means that if we know two angles in a triangle, we can always work out the third one.

For similarity, this means that we also just need to check two angles to determine if triangles are similar. If two triangles have two angles of the same size, then the third angle must also be the same in both.

This result is sometimes called the AA Similarity Condition for triangles. (The two As stand for the two angles we compare.)

If two angles in one triangle are congruent to two angles in another triangle, the two triangles are similar.

Let’s have a look at a few examples where this is useful:

COMING SOON – Animation

Here you can see the image of a large lighthouse. Together with a friend, you want to measure the height of the lighthouse, but unfortunately we cannot climb to the top.

It turns out that, very well hidden, the diagram contains two similar triangles: one is formed by the lighthouse and its shadow, and one is formed by your friend and her shadow.

Both triangles have one right angle at the bottom. The sun rays are parallel, which means that the other two angles at the bottom are corresponding angles, and also equal. By the AA condition for triangles, these two must be similar.

We can easily measure the length of the shadows, and we also know the height of your friend. Now we can use the proportionality of sides in similar triangles to find the height of the lighthouse:

COMING SOON – Equation

Therefore the lighthouse is 1.5m tall.

COMING SOON – Animation

We can use the same technique to measure distances on the ground. Here we want to find the width of a large river. There is a big tree on one side of the river, and I’ve got a stick that is one meter long.

Try drawing another two similar triangles in this diagram.

You can mark the point along the side of the river, that lies directly on the line of sight from the end of the stick to the tree. Then we can measure the distances to the stick, and to the point directly opposite the tree.

Once again, these two triangles are similar because of the AA condition. They both have a right angle, and on pair of opposite angles.

According to the proportionality rule, this means that

COMING SOON – Equation

Therefore the width of the river is 45 meters.

Similarity on Rays

Theorem: If a ray bisects an angle of a triangle, then it divides the opposite side into segments that are proportional to the lengths of the other two sides.

We can extend this theorem to a situation outside of triangles where we have multiple parallel lines cut by transverals.

Theorem: If three or more parallel lines are cut by two transversals, then they divide the transversals proportionally.

Think about a midsegment of a triangle. A midsegment is parallel to one side of a triangle and divides the other two sides into congruent halves. The midsegment divides those two sides proportionally.

Triangle Proportionality Theorem: If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally.

Triangle Proportionality Theorem Converse: If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

Self Similarity

There are some curious mathematical shapes that are similar to a smaller part of themselves. An example is the Sierpinksi Triangle: the entire triangle is similar to any one of the smaller triangles it consists on. You could zoom in and infinitely many smaller and smaller triangles.

Shapes with this property are called Fractals. They have some surprising and truly XXX properties, which you will learn about more in the future.

Triangles are not just useful for measuring distances. In the next course we will learn a lot more about triangles and their properties.

Sign in to Innings2