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Similar Figures

Two figures are said to be similar if they are the same shape. In more mathematical language, two figures are similar if their corresponding angles are congruent , and the ratios of the lengths of their corresponding sides are equal.

This common ratio is called the scale factor .

The symbol is used to indicate similarity.

Example 1:

In the figure below, pentagon A B C D E pentagon V W X Y Z .

(Note that the order in which you write the vertices matters; for instance, pentagon A B C D E is not similar to pentagon V Z Y X W .)

Example 2:

The two cylinders are similar. Find the scale factor and the radius of the second cylinder.

The height of the cylinder on the right is 1 3 the height of the cylinder on the left. So, the scale factor is 1 3 .

To get the radius of the smaller cylinder, divide 1.8 by 3 .

1.8 ÷ 3 = 0.6

So, the radius of the smaller cylinder is 0.6 cm.

Note that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations , reflections , translations , and dilations .

Example 3:

In the figure above, the hexagon A 1 B 1 C 1 D 1 E 1 F 1 is flipped horizontally to get A 2 B 2 C 2 D 2 E 2 F 2 .

Then hexagon A 2 B 2 C 2 D 2 E 2 F 2 is translated to get A 3 B 3 C 3 D 3 E 3 F 3 .

Hexagon A 3 B 3 C 3 D 3 E 3 F 3 is dilated by a scale factor of 1 2 to get A 4 B 4 C 4 D 4 E 4 F .

Note that

A 1 B 1 C 1 D 1 E 1 F 1 A 2 B 2 C 2 D 2 E 2 F 2 A 3 B 3 C 3 D 3 E 3 F 3 A 4 B 4 C 4 D 4 E 4 F 4 .

That is, all four hexagons are similar. (In fact, the first three are congruent.)

Example 4:

Consider pentagon P Q R S T on a coordinate plane.

A rotation by 180 ° about the origin takes the pentagon to P ' Q ' R ' S ' T ' .

Now, a dilation about the origin by a scale factor 2 takes the pentagon P ' Q ' R ' S ' T ' to P ' ' Q ' ' R ' ' S ' ' T ' ' .

Note that P Q R S T P ' Q ' R ' S ' T ' P ' ' Q ' ' R ' ' S ' ' T ' ' . That is, all three pentagons are similar. (And the first two are congruent.)