Geometry (A): READ: Similar Polygons Reading

Similar Polygons

Learning Objectives

Recognize similar polygons.
Identify corresponding angles and sides of similar polygons from a statement of similarity.
Calculate and apply scale factors.

Introduction

Similar figures, rectangles, triangles, etc., have the same shape. Same shape, however, is not a precise enough term for geometry. In this lesson, we’ll learn a precise definition for similar, and apply it to measures of the sides and angles of similar polygons.

Similar Polygons

Look at the triangles below.

The triangles on the left are not similar because they are not the same shape.
The triangles in the middle are similar. They are all the same shape, no matter what their sizes.
The triangles on the right are similar. They are all the same shape, no matter how they are turned or what their sizes.

Look at the quadrilaterals below.

The quadrilaterals in the upper left are not similar because they are not the same shape.
The quadrilaterals in the upper right are similar. They are all the same shape, no matter what their sizes.
The quadrilaterals in the lower left are similar. They are all the same shape, no matter how they are turned or what their sizes.

Now let’s get serious about what it means for figures to be similar. The rectangles below are all similar to each other.

These rectangles are similar, but it’s not just because they’re rectangles. Being rectangles guarantees that these figures all have congruent angles. But that’s not enough. You’ve seen lots of rectangles before, some are long and narrow, others are more blocky and closer to square in shape.

The rectangles above are all the same shape. To convince yourself of this you could measure the length and width of each rectangle. Each rectangle has a length that is exactly twice its width. So the ratio of length-to-width is $2:1$ for each rectangle. Now we can make a more formal statement of what similar means in geometry.

Two polygons are similar if and only if:

they have the same number of sides
for each angle in either polygon there is a corresponding angle in the other polygon that is congruent
the lengths of all corresponding sides in the polygons are proportional

Reminder: Just as we did with congruent figures, we name similar polygons according to corresponding parts. The symbol $\sim$ is used to represent “is similar to.” Some people call this “the congruent sign without the equals part.”

Example 1

Suppose $\triangle{ABC}\sim \triangle{JKL}$ Based on this statement, which angles are congruent and which sides are proportional? Write true congruence statements and proportions.

$\angle{A} \cong \angle{J}, \angle{B} \cong \angle{K},$ and $\angle{C} \cong \angle{L}$

$\frac{AB} {JK} = \frac{BC} {KL} = \frac{AC} {JL}$

Remember that there are many equivalent ways to write a proportion. The answer above is not the only set of true proportions you can create based on the given similarity statement. Can you think of others?

Example 2

Given: $MNPQ \sim RSTU$

What are the values of $x, y,$ and $z$ in the diagram below?

Set up a proportion to solve for $x$ :

$\frac{x} {25} &= \frac{18} {30} \\ \frac{x} {25} &= \frac{3} {5} \\ 5x &= 75 \\ x &= 15$

Now set up a proportion to solve for $y$ :

$\frac{y} {15} &= \frac{30} {18} \\ \frac{y} {15} &= \frac{5} {3} \\ 3y &= 75 \\ y &= 25$

Finally, since $Z$ is an angle, we are looking for $m\angle{R}$

$Z = m\angle {R}=m\angle {M} = 115^\circ$

Example 3

$ABCD$ is a rectangle with length $12$ and width $8$ .

$UVWX$ is a rectangle with length $24$ and width $18$ .

A. Are corresponding angles in the rectangles congruent?

Yes. Since both are rectangles, all the angles in both are congruent right angles.

B. Are the lengths of the sides of the rectangles proportional?

No. The ratio of the lengths is $12:24 = 1:2$ . The ratio of the widths is $8:18 = 4:9 \ne 1:2$ . Therefore, the lengths of the sides are not proportional.

C. Are the rectangles similar?

No. Corresponding angles are congruent, but lengths of corresponding sides are not proportional.

Example 4

Prove that all squares are similar.

Our proof is a “paragraph” proof in bullet form, rather than a two-column proof:

Given two squares.

All the angles of both squares are right angles, so all angles of both squares are congruent—and this includes corresponding angles.
Let the length of each side of one square be $k$ , and the length of each side of the other square be $m$ . Then the ratio of the length of any side of the first square to the length of any side of the second square is $k: m$ . So the lengths of the sides are proportional.
The squares satisfy the definition of similar polygons: congruent angles and proportional side lengths - so they are similar

Scale Factors

If two polygons are similar, we know that the lengths of corresponding sides are proportional. If $k$ is the length of a side in one polygon, and $m$ is the length of the corresponding side in the other polygon, then the ratio $\frac{k} {m}$ is called the scale factor relating the first polygon to the second. Another way to say this is:

The length of every side of the first polygon is $\frac{k} {m}$ times the length of the corresponding side of the other polygon.

Example 5

Look at the diagram below, where $ABCD$ and $AMNP$ are similar rectangles.

A. What is the scale factor?

Since $ABCD \sim AMNP$ , then $AM$ and $AB$ are corresponding sides. Since $ABCD$ is a rectangle, you know that $AB = DC = 45.$

The scale factor is the ratio of the lengths of any two corresponding sides.

So the scale factor (relating $ABCD$ to $AMNP$ ) is $\frac{45} {30} = \frac{3} {2} = 1.5$ . We now know that the length of each side of $ABCD$ is $1.5\;\mathrm{times}$ the length of the corresponding side in $AMNP$ .

Comment: We can turn this relationship around “backwards” and talk about the scale factor relating $AMNP$ to $ABCD$ . This scale factor is just $\frac{30} {45} = \frac{2} {3},$ which is the reciprocal of the scale factor relating $ABCD$ to $AMNP.$

B. What is the ratio of the perimeters of the rectangles?

$ABCD$ is a $45$ by $60$ rectangle. Its perimeter is $45 + 60 + 45 + 60 = 210$ .

$AMNP$ is a $30$ by $40$ rectangle. Its perimeter is $30 + 40 + 30 + 40 = 140$ .

The ratio of the perimeters of $ABCD$ to $AMNP$ is $\frac{210} {140} = \frac{3} {2}$ .

Comment: You see from this example that the ratio of the perimeters of the rectangles is the same as the scale factor. This relationship for the perimeters holds true in general for any similar polygons.

Ratio of Perimeters of Similar Polygons

Let’s prove the theorem that was suggested by example 5.

Ratio of the Perimeters of Similar Polygons:

If $P$ and $Q$ are two similar polygons, each with $n$ sides and the scale factor of the polygons is $s$ , then the ratio of the perimeters of the polygons is $s$ .

Given: $P$ and $Q$ are two similar polygons, each with $n$ sides

The scale factor of the polygons is $s$

Prove: The ratio of the perimeters of the polygons is $s$

Statement	Reason
1. $P$ and $Q$ are similar polygons, each with $n$ sides	1. Given
2. The scale factor of the polygons is $s$	2. Given
3. Let $p_1 , p_2 ,\ldots, p_n$ and $q_1 , q_2 ,\ldots, q_n$ be the lengths of corresponding sides of $P$ and $Q$	3. Given (polygons have $n$ sides each)
4. $p_1 = sq_1,\;p_2 = sq_2 ,\;\ldots\;, p_n = sq_n$	4. Definition of scale factor
5. Perimeter of $P = p_1 + p_2 + \ldots + p_n$	5. Definition of perimeter
6. $= sq_1 + sq_2 + \ldots + sq_n$	6. Substitution
7. $= s \left(q_1 + q_2 + \ldots + q_n\right)$	7. Distributive Property
8. $= s$ , the perimeter of $Q$	8. Definition of perimeter $\blacklozenge$

Comment: The ratio of the perimeters of any two similar polygons is the same as the scale factor. In fact, the ratio of any two corresponding linear measures in similar figures is the same as the scale factor. This applies to corresponding sides, perimeters, diagonals, medians, midsegments, altitudes, etc.

As we’ll see in an upcoming lesson, this is definitely not true for the areas of similar polygons. The ratio of the areas of similar polygons (that are not congruent) is not the same as the scale factor.

Example 6

$\triangle{ABC}\sim\triangle{MNP}$ . The perimeter of $\triangle{MNP}$ is $150.$

What is the perimeter of $\triangle{ABC}$ ?

The scale factor relating $\triangle{ABC}$ to $\triangle{MNP}$ is $\frac{32} {48} = \frac{2} {3}$ . According to the Ratio of the Perimeter's Theorem, the perimeter of $\triangle{ABC}$ is $\frac{2} {3}$ of the perimeter of $\triangle{MNP}$ . Thus, the perimeter of $\triangle{ABC}$ is $\frac{2} {3} \cdot 150 = 100$ .

Lesson Summary

Similar has a very specific meaning in geometry. Polygons are similar if and only if the lengths of their sides are proportional and corresponding angles are congruent. This is same shape translated into geometric terms.

The ratio of the lengths of corresponding sides in similar polygons is called the scale factor. Lengths of other corresponding linear measures, such as perimeter, diagonals, etc. have the same scale factor.

Points to Consider

Scale factors show the relationship between corresponding linear measures in similar polygons. The story is not quite that simple for the relationship between the areas or volumes of similar polygons and polyhedra (three-dimensional figures). We’ll study these relationships in future lessons.

Similar triangles are the basis for the study of trigonometry. The fact that the ratios of the lengths of corresponding sides in right triangles depends only on the measure of an angle, not on the size of the triangle, makes trigonometric functions the property of an angle, as you will study in Chapter 8.

The following questions are for your own review. The answers are listed below for you to check your work and understanding.

Review Questions

True or false?

All equilateral triangles are similar.
All isosceles triangles are similar.
All rectangles are similar.
All rhombuses are similar.
All squares are similar.
All congruent polygons are similar.
All similar polygons are congruent.

Use the following diagram for exercises 8-11.

Given that rectangle $ABCD:$ rectangle $AMNP$ .

What is the value of each expression?

$AB$
$BC$
$MB$
$PD$
Given that $\triangle ABC \cong \triangle MNP$ , what is the scale factor of the triangles?

Use the diagram below for exercises 13-16.

Given: $\overline{PQ}: \overline{ST}$

What is the perimeter of $\triangle {PQR}?$
What is the perimeter of $\triangle {TSR}?$
What is the ratio of the perimeter of $\triangle {PQR}$ to the perimeter of $\triangle {TSR}?$
Prove: $:PQR: TSR$ . [Write a flow proof.]
is the midpoint of and is the midpoint of in .
1. Name a pair of parallel segments.
2. Name two pairs of congruent angles.
3. Write a statement of similarity of two triangles.
4. If the perimeter of the larger triangle in $c$ is $p$ , what is the perimeter of the smaller triangle?
5. If the area of $\triangle {ABC}$ is $100$ , what is the area of quadrilateral $AMNC?$

Review Answers

True
False
False
False
True
True
False
$45$
$60$
$15$
$20$
$1: 1, 1,$ or $1.0$
$8$
$16$
$1: 2, \frac{1} {2}, 0.5$ or equivalent
$PQ: TS = QR: SR = PR: TR = 1:2$ , so the sides are all proportional.
$& \angle{PRQ} \cong \angle{TSR} && \text{(vertical angles) }\\ & \angle{RPQ} \cong \angle{RTS}, \angle{RQP} \cong \angle{RST} && \text{(parallel lines, alternate interior angles are congruent)}\\ & \triangle{PQR}: \triangle{TSR} && \text{(definition of similar polygons: angles are congruent},\\ &&& \text{lengths of sides are proportional)}$
1. $\overline{MN}, \overline{AC}$
2. $\angle{BMN} \cong \angle{BAC} \angle{BNM} \cong \angle{BCA}$
3. $\triangle{BAC}: \triangle{BMN}$
4. $\frac{1} {2}p, \frac{p} {2}$ or equivalent
5. $75$

Last modified: Monday, June 28, 2010, 6:11 PM