Similarity in Geometry: Application & Practice

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson will go over similarity in geometry. We will be defining similar shapes, looking at their properties, and using these properties in real-world, everyday applications.

Similarity

It's moving day!!! Well, hypothetically speaking, anyway! Suppose you are moving into a new place, and the day has finally come that you have to pack up your old place and move out! Before you start packing, you go to the local moving supplies store to buy some moving boxes. All of the boxes are unassembled, so they look like cardboard rectangles. The boxes come in sizes extra small, small, medium, and large.


simgeom1


As you are deciding which boxes to buy, you notice that all of the boxes have the exact same shape, but they differ in size. This rings a bell, and you remember that last week in math class, your teacher was talking about something called similarity in geometry. You try to remember what it was that he said about this subject.

Oh yeah! You remember that, in geometry, similar objects, shapes, or figures have the same shape but different sizes. Well, no wonder this popped into your head when observing the moving boxes! You realize that all of the boxes are similar rectangles!

Properties of Similar Shapes

As we said, when two shapes are similar, they have the same shape, but differ in size. In other words, we can obtain one shape from the other by resizing one of the shapes. Because of this, similar shapes have two important properties that have to do with the measures of their angles and the lengths of their sides.

  1. In similar shapes, corresponding angles have the same measure.
  2. In similar shapes, the lengths of corresponding sides are proportional.

To illustrate this, consider an extra small box and a large box at the moving supplies store.


simgeom2


We see that corresponding angles of the boxes are equal and corresponding sides are proportional.

These properties are extremely useful in applications involving similar shapes, objects, or figures. Let's take a look at some examples to illustrate this!

Applications

You're still trying to figure out which boxes to buy for the move. You notice that the lengths of the sides of the small box are indicated on the box, but the medium box only has the length of one side indicated.


simgeom3


Hmmm, if you want to know specific measurements, such as the area of the flattened boxes, you will need to know the length of the other side on the medium box! Well, guess what? We're about to see just how useful those properties of similar shapes can be! We know that the two flattened boxes are similar rectangles. Because of this, we know that the lengths of their corresponding sides are proportional.

If we let the unknown side length be x, we can use this property to set up an equation in x, and solve, giving us the unknown side length! We have that:

AB = 2ft.

AD = 3ft.

EF = 4ft.

EH = x

We also know that (AB / EF) = (AD / EH). Plugging in the different lengths in the equation, we get an equation in x.


simgeom4


We see that (2 / 4) = (3 / x). Now, we simply solve for x.


simgeom5


We see that x = 6, so we know that the side with the missing length on the medium box is 6 feet long, and it's all thanks to the properties of similar shapes!

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