## Similar figures and proportions

Definition: Similar figures are figures that are the exact same shape, but are different sizes.

 Similar figures Not similar figures

Two figures are similar if the lengths of their corresponding sides form a proportion.

Example 1

These two triangles are similar. We can prove that they are similar using a ratio table to compare the lengths of their corresponding sides.

 Big triangle Small triangle 20 15 8 6 16 12
• In the table, you can see that the ratio of the bases is 15/20. This reduces to $\frac{3}{4}$.
• Also in the table you can see that the ratio of the two right sides is 6/8. This reduces to $\frac{3}{4}$.
• Finally, the ratio of the two left sides is 12/16. This reduces to $\frac{3}{4}$.

Since all three ratios are equivalent, then the two triangles are similar.

 $\frac{15}{20}=\frac{6}{8}=\frac{12}{16}$ Therefore, the two triangles are similar.

 Directions: Use the "shapeswitcher" to choose the similar figures. Move the vertices, sides, and figures themselves. Notice that the lengths change, but the two figures maintain their similarity. Use the "Show/Hide ratios" button to verify that the ratios are indeed equivalent.

Example 2

These two triangles are similar. Find the missing length.

To find the missing length we need to create a ratio table, like so...

 Big triangle Small triangle 10 6 25

Once the table is made, it is up to you to figure out this missing value. In this case, the missing value is 15 cm. Can you see how to get that answer?

Need a hint? [show the hint]

 Directions: Move the vertices of the similar triangles. Use proportions to find the missing lengths. Pull down the sliders to check your answers.

# Self-Check

 Question 1 These two triangles are similar. Find the missing length. [show answer]   $15\,\mbox{cm}$

 Question 2 These two rectangles are the same shape, but one has been scaled up in size. What is the missing length of the larger rectangle‌ [show answer]   $9.6\mbox{ cm or }9\frac{3}{5}\mbox{ cm }$

 Question 3 Suppose a triangle is placed into a photocopier machine and a copy is made that has been reduced in size by some scale factor. Find the length of y. [show answer]   $y\,=\,16\mbox{ cm}$