What are Similar Figures?
Similar figures are figures that have the same shape. They can be the same exact size, or one can be larger than the other.
The symbol for "is similar to" is a little squiggly line (called a tilde).
When two shapes are similar, their corresponding angles will be the same. The angles are the same because the shape is still the same. You can tell from the order of the similarity statement which angles match up. In the example, angles A and D are the same (both letters are in written first), angles B and E are the same (both letters are written in the middle), and C and F are the same (both letters are written last).
Similar triangles can be different sizes, so the lengths of the corresponding sides are not necessarily the same, but they are proportional. This means that the ratios of the corresponding sides are equal.
This might be easier to understand with an example. If we divide all the corresponding sides (in the same order), we see that each ratio came out to 3/4. When two shapes are similar, the ratios of the corresponding sides will be the same. In other words, the side lengths are proportional. Not sure what a proportion is? Check out the lesson on proportions.
Open the next tab to learn about scale factors of similar figures.
What is a Scale Factor?
The ratio that you get when you divide corresponding side lengths of similar figures is called the scale factor. In the last example, the ratios all simplified to 3/4 so we would say that the scale factor of triangle QRS to triangle LMN is 3/4.
Another way to describe a scale factor is that it's a multiplier. In the example below, the scale factor of triangle ABC to triangle DEF is 2. This means that the second triangle is 2 times as big. If you multiply a side from triangle ABC by 2, you get the length of the corresponding side of triangle DEF. You can also get 2 as the scale factor by finding the ratios: 12/6 = 2, 16/8 = 2, and 18/9 = 2. The ratios of the corresponding sides are all equal to 2.
Here's another example. The scale factor of rectangle LMNO to rectangle STUV is 3. That means that the second rectangle is 3 times as big as the first rectangle. If you multiply the length of a side of the first rectangle by 3, you get the length of the corresponding side of the second rectangle.
You can also find the scale factor for the rectangles above by finding the ratios. 15/5 = 3 and 6/2 = 3. If you can't figure out what the side of the first shape is being multiplied by, you can go backwards and divide to find the scale factor.
Does the Order Matter?
Yes. The order is very important when you write a scale factor. If the scale factor of shape #1 to shape #2 is 4, that means the second shape is 4 times as big. If you look at the ratios, 20/5 = 4 and 16/4 = 4.
But what happens if we switch the order? We wouldn't want to say that the scale factor is still 4 if the second shape is smaller. Remember, the scale factor is a multiplier. What can we multiply by if the second shape is smaller?
If you know the scale factor when the second shape is bigger, the scale factor of the other way around will be its reciprocal (flip the fraction upside down). This happens because the ratios will all be switched around. In the example above, we 20/5 and 16/4 to get the scale factor when the second shape was larger. If we switch the order, the ratios become 5/20 and 4/16, which both reduce to 1/4.
Pay close attention to the order of the two shapes. The first shape listed is what you start with. If the second shape is larger, the scale factor will be greater than 1. If the second shape is smaller, the scale factor will be smaller than 1.
Open the next tab to see a few more examples.
Assume all shapes below are similar. Pay careful attention to the order of the shapes when finding the scale factor.
Find the scale factor of trapezoid EFGH to trapezoid JKLM.
Notice that the smaller shape was listed first. If the second shape is larger, that means the scale factor will be larger than 1. What can you multiply a side of the first trapezoid by to get the length of the corresponding side on the larger trapezoid? If you multiply each side of the first trapezoid by 2, you get the lengths of the sides from the larger trapezoid. This means the scale factor is 2. You can also see that it's two by finding the ratios: 14/7 = 2, 10/5 = 2, 16/8 = 2.
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