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When working with transformations, remember that the composition of rigid motions is a rigid motion. However, since a dilation is not a rigid motion, what happens if a rigid motion and a dilation are composed? The answer to that question will be developed throughout this lesson.

Catch-Up and Review

Here is some recommended reading before getting started with this lesson.

Concept and properties of dilation.
Concept and properties of rigid motion.
Concept and properties of congruence.

Discussion

Properties of a Similarity Transformation

Previously, it was seen that rigid motions keep the figure's size and shape. In comparison, dilations keep the figure's shape but can change its size. The next natural question is, what does a similarity transformation do to a figure?

The following is a list of a few important properties of similarity transformations.

The image of a line segment is a line segment. The image is longer or shorter than the preimage in the ratio given by the scale factor.
Similarity transformations preserve angle measures.

Proof

These are properties of both rigid motions and dilations. Since similarity transformations are combinations of rigid motions and dilations, these are also properties of similarity transformations.

Discussion

Criteria for Similar Polygons

For polygons, similarity can be checked by considering angle measures and side lengths.

Example

Investigating Similarity of Different Polygons

Determine whether the following statements are true or false.

a All quadrilaterals are similar.

Hint

Do all quadrilaterals have the same angles?

Solution

Different quadrilaterals may have different angles. Since similar polygons have congruent corresponding angles, this means that not all quadrilaterals are similar.

b All trapezoids are similar.

Hint

Do all trapezoids have the same angles?

Solution

Different trapezoids may have different angles. Since similar polygons have congruent corresponding angles, this means that not all trapezoids are similar.

c All parallelograms are similar.

Hint

Do all parallelograms have the same angles?

Solution

Different parallelograms may have different angles. Since similar polygons have congruent corresponding angles, this means that not all parallelograms are similar.

d All rhombi are similar.

Hint

Do all rhombi have the same angles?

Solution

Different rhombi may have different angles. Since similar polygons have congruent corresponding angles, this means that not all rhombi are similar.

e All rectangles are similar.

Hint

Consider the ratio of the sides.

Solution

The corresponding sides of similar polygons are proportional. Since $1 : 2 \neq = 3 : 4,$ the rectangles on the diagram are not similar.

Not all rectangles are similar.

f All squares are similar.

Hint

Do all squares have the same shape?

Solution

Consider the angles and the sides of a square with side length $m$ and a square with side length $n .$

All angles of both squares are right angles, so the corresponding angles of the two squares are congruent.
The ratio between any sides of the two squares is $m : n,$ so the corresponding sides are proportional.

These two properties guarantee that the squares are similar.

Any two squares are similar.

Example

Solving Problems Using Similarity of Polygons

The triangles on the diagram are similar. It is given that the length of

A B

is

4

centimeters, the length of

B C

is

3

centimeters, and the length of

A C

is

5

centimeters.

△ A B C \sim △ A C D \sim △ A D E \sim △ A E F \sim △ A F G

Find

A G .

Give your answer rounded to the nearest millimeter.

There are five triangles joined by their sides: ABC, ACD, ADE, AEF, and AFG.

Hint

Find the length of $\overline{A D}$ first.

Solution

Focus on the first two triangles to find the length of $\overline{A D}$ first.

It is given that

△ A B C

is similar to

△ A C D,

so the corresponding sides are proportional.

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

The lengths of

\overline{C A}

and

\overline{B A}

are given in the question. Substituting these values in the equation gives the length of

\overline{D A} .

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

$C A = 5$ , $B A = 4$

\frac{D A}{5} = \frac{5}{4}

$LHS \cdot 5 = RHS \cdot 5$

D A = \frac{5}{4} \cdot 5

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

D A = \frac{2 5}{4}

Similar argument gives the length of

\overline{E A},

\overline{F A},

and

\overline{G A} .

The other three triangles are also similar to $△ A B C,$ so the corresponding sides are proportional.

Proportion		Solution
Expression	Substitution	Solution
$\frac{E A}{C A} = \frac{D A}{B A}$	$\frac{E A}{5} = \frac{2 5 / 4}{4}$	$E A = \frac{2 5 / 4}{4} \cdot 5 = \frac{1 2 5}{1 6}$
$\frac{F A}{C A} = \frac{E A}{B A}$	$\frac{F A}{5} = \frac{1 2 5 / 1 6}{4}$	$F A = \frac{1 2 5 / 1 6}{4} \cdot 5 = \frac{6 2 5}{6 4}$
$\frac{G A}{C A} = \frac{F A}{B A}$	$\frac{G A}{5} = \frac{6 2 5 / 6 4}{4}$	$G A = \frac{6 2 5 / 6 4}{4} \cdot 5 = \frac{3 1 2 5}{2 5 6}$

The length of $\overline{G A}$ is $\frac{3 1 2 5}{2 5 6},$ or approximately $12.2$ centimeters.

Pop Quiz

Practice Using Similarity to Solve Problems

On the diagram all quadrilaterals are similar.

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