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What Is the Definition of Similar Figures?

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Similar triangles

Theory

Similarity

Two shapes are similar if you can get one by scaling the other. In other words, they have the same shape, but not necessarily the same size.

You have similar triangles if one or more of these conditions are satisfied:

Rule

Requirements of Similar Triangles

Two corresponding angles in each triangle are equal.
The ratios of all corresponding sides are equal.
The ratios of two corresponding sides are equal, and the corresponding angles between them are equal.
The ratios of two corresponding sides in each triangle are equal, and the opposite angles to the longest of them are equal.

Example 1

You are given two triangles $△ A B C$ and $△ D E F$ . You know that $A B = 4$ , $B C = 5$ , $C A = 8$ , and that $D E = 12$ , $E F = 15$ and $F D = 24$ . You can determine that these triangles are similar, because

\frac{D E}{A B} = \frac{E F}{B C} = \frac{F D}{C A} = 3

You also see from this calculation that the ratio of the sides is 3.

Note! Feel free to draw a figure. It is very helpful!

Above, you learned how to determine if two triangles are similar. Now you will learn how to use similarity to determine the length of a side in a triangle.

When you have two triangles that are similar, then you know that the ratio between the sides is the same in both triangles. By comparing corresponding sides in the triangles, you can find the length of a side.

Similarity of triangles

The triangles in the figure above are similar.

Now, set up two fractions with an equal sign between them, where the numerator in each fraction is a corresponding side from each shape, and the denominators are also two corresponding sides:

\frac{B C}{A B} = \frac{E F}{D E}

You can solve this as an equation for the side you want to find the length of. It is important that you make sure you compare corresponding sides. This means that you may have to turn the triangle a little to see which sides actually correspond to each other.

Example 2

The triangles $△ A B C$ and $△ D E F$ are similar.

Find the length of $B C$ .

\begin{array}{l} \frac{B C}{A C} & = \frac{E F}{D F} \\ \frac{B C}{2} & = \frac{5}{3} \\ B C & = \frac{10}{3} \end{array}

Example 3

The triangles $△ A B C$ and $△ D E F$ are similar.

Find the length of $D E$ .

\begin{array}{l} \frac{D E}{E F} & = \frac{A B}{B C} \\ \frac{D E}{3} & = \frac{6}{2} \\ D E & = \frac{6}{2} \cdot 3 \\ D E & = 9 \end{array}

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How Do You Find the Ratio of Similarity?

How Do You Know if Triangles Are Congruent?

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