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Similarity in Geometric Shapes

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  • 1:46 The Test
  • 4:19 Comparing Two Shapes
  • 5:16 Lesson Summary
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Instructor
Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Expert Contributor
Simona Hodis

Simona received her PhD in Applied Mathematics in 2010 and is a college professor teaching undergraduate mathematics courses.

View this lesson to learn how you can determine whether shapes are similar. Find out how to use the one rule that tells you when your two shapes are geometrically similar.

Geometric Shapes

In this lesson, we'll look at geometric shapes. When we say 'geometric shape,' we're talking about the form or shape of a particular object. For example, a ball is round in shape. Most often, we talk about two-dimensional shapes, such as the ones you draw on paper, like circles, triangles, and rectangles. The very first house I drew as a kid consisted of a square for the building of the house, a triangle for the roof, a rectangle for the door, and small squares for the windows. You might have drawn something similar. All we did was put together different geometric shapes to create a picture.

Similar Shapes

Since then, I've learned to draw different types of houses and houses of different sizes. When I drew the houses that are different sizes, I kept the house looking the same, but I just drew the houses either larger or smaller. When I did this, I actually drew similar shapes.

We can define similar shapes as shapes that are the same with corresponding sides that are proportional in size. What does this mean? It sounds very technical, but it isn't. Why would you want to learn about similar shapes? Knowing how to work with similar shapes is a good skill to have not only for when you take math tests, but also in real life.

Architects and toy designers, for example, use similar shapes all the time to make different-sized version of things. A toy designer might want to create a miniature house. To do this, the toy designer will need to make sure that all the shapes in the miniature house are similar to the shapes in a life-size house. For the miniature house to look just like the real house, all the shapes must be similar. Yes, the shapes will look the same, except that their size is smaller in the miniature house and larger in the real house.

The Test

Because similar shapes differ only in size, there is a test we can perform to make sure that our shapes are really similar. See, sometimes, when you're given a drawing of two shapes, they may not be exactly similar. For example, if you were the toy designer making the miniature house and a coworker comes and gives you a tiny window for you to use, you would want to check to make sure that the window is a similar shape to the real-world window that you are making a tiny version of. If the tiny window is not a similar shape, then you wouldn't be able to use it because it wouldn't look the same. You wouldn't have a real miniature of the house.

Let's take a look at what the test involves. You have drawn the window your coworker gave you next to the real window. You have marked the dimensions of both.

similar shapes

Now, the test tells you that the two shapes are similar when all corresponding sides have the same ratio of proportionality. This means that for each corresponding side, when we take the ratio of one side to the other side, the ratio will be same for all corresponding sides of the two shapes.

So, in your case, for the windows, your ratios are 2/16 for the shorter sides and 3/24 for the longer sides. We matched the sides with each other and made sure to write our ratios so that the first shape is always the top number. You can write your ratios any way you like, but you have to be consistent. If your second shape is on the top, then it must be on the top for all the other ratios.

So for our two windows to be similar, the ratios 2/16 and 3/24 must be the same. At first glance, you might say that they are not similar. But hold on a minute; you can't say anything until you've simplified your ratios. Yes, just like when you're working with fractions, your numbers must be simplified before you're done.

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Additional Activities

Similarity in Geometric Shapes

Review of the Topic

We say that two geometric shapes are similar if their side lengths are proportional. In other words, if we scale up or down the two shapes (together with a translation, rotation, or reflection), they perfectly overlap.

To determine if two shapes are similar, knowing all their side lengths, we will verify if the sides are proportional.

However, if not all sides are known, but some sides and certain angles of the shapes are, we can determine if they are similar by checking the following properties.

Similar Triangles

To check if two triangles are similar, we have the following properties:

  1. (SSS) If all the corresponding sides are proportional then the triangles are similar.
  2. (AA) If two of the corresponding angles are congruent, the triangles are similar.
  3. (SAS) If two corresponding side lengths are proportional and the angle between the two sides has the same measure, then the triangles are similar.

Applications

In the following questions, state which of the similarity property from SSS, AA, SAS are satisfied, write the proportions of the sides, and answer the questions.

1. Find x and y if BD = 4, ED = 7, BE = 5, AE = 3 and the angle D is equal to the angle A as marked in the figure.

2. Find the ratio of x and y if AB=5, BC = 4, DF = 5/4, DE = 1, and the angles B and D are the same.

Solutions

1. The similarity property is AA because angles D and A are equal and angle B is common.

Therefore, x = 6 and y = 14.

2. The similarity property is SAS because angles B and D are equal, and the sides AB, DC, and DF, DE are proportional.

And the ratio is x = 4y.

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