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3. Symmetries of Plane Figures

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Chapter 1

Symmetries of Plane Figures

This lesson explores the fascinating world of symmetries in plane figures, specifically focusing on parallelograms and trapezoids. It delves into two types of symmetry: line and rotational. For instance, a parallelogram has rotational symmetry, meaning it can be rotated to match its original position. On the other hand, an isosceles trapezoid has line symmetry, which means it can be folded along a line to create two matching halves. These concepts are not just theoretical; they have practical applications too. Architects and designers often use these symmetries when creating blueprints or models. Understanding these symmetries can also be crucial in fields like robotics and computer graphics, where precise calculations and designs are essential.

Lesson Settings & Tools

	16 Theory slides
	8 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Image Credits

Slide of 16

Transformations can change the size, position, or orientation of a figure. They can also map one figure onto another. But, can a transformation map a figure onto the same figure? Throughout the lesson, the answer to this question will be developed.

Catch-Up and Review

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

Quadrilateral
Regular polygon
Transformations

Explore

Rotational Symmetry in Rectangles

Consider a rectangle

A B C D .

Can the rectangle be rotated such that the image exactly matches the preimage? Not so fast, there is one catch; can it be done where the angle of rotation is greater than

0^{\circ}

and less than

36 0^{\circ} ?

Try it out by rotating

A B C D

around the movable point

P .

If accomplished, where was point

P

located? What was the angle of rotation?

Explore

Line Symmetry in Squares

In the applet, square

A B C D

can be reflected along line

ℓ .

Is it possible for the image of

A B C D

under a reflection to match the preimage exactly? Try it out!

Applet to reflect square ABCD along different lines

If it was possible, along which points of

A B C D

did the line pass? Could there be more than one line that works?

Discussion

Defining Symmetry

For some geometric figures, it is possible to find a transformation that maps the figures onto themselves. In such cases, the transformation is called a symmetry of the figure.

Concept

Symmetry

A symmetry is a rigid motion that maps a figure onto itself. It is a non-trivial congruence of a figure with itself, that is, the sides and angles do not correspond with themselves. When a figure has a symmetry it is said to be a symmetric figure.

Discussion

Defining Rotational Symmetry

When a figure is rotated less than $36 0^{\circ},$ the final image can look the same as the initial one — as if the rotation did nothing to the preimage. In such a case, the figure is said to have rotational symmetry.

Example

Identifying Rotational Symmetry

While walking downtown, Heichi and Paulina saw a store with the following logo. They began to discuss whether the logo has rotational symmetry. To figure it out, they went into the store and took a business card each.

At their respective homes, and using patty paper, each performed rotations on the logo. The same night, Heichi called Paulina to tell her that the logo is rotationally symmetric, but Paulina disagreed. Who is correct?

In the event that the figure has rotational symmetry, what is the order of symmetry?

Hint

Rotate the logo $36 0^{\circ}$ about its center. If possible, verify where along the way the rotation matches the original logo.

Solution

To determine whether the logo has rotational symmetry, check for a rotation of less than

36 0^{\circ}

about the center that maps the logo onto itself. By the use of patty paper or the following applet, rotational symmetry can be checked.

As shown, a rotation of

12 0^{\circ}

maps the logo onto itself. Therefore, the logo is rotationally symmetric, which means Heichi was right. Additionally, two other rotations of

24 0^{\circ}

and

36 0^{\circ}

also map the logo onto itself. This means that the order of symmetry is

3 .

Example

Identifying Rotational Symmetry in Parallelograms

In the first exploratory applet, it was studied whether a rectangle has rotational symmetry. Now, a parallelogram will be studied. Consider a parallelogram $A B C D .$

Does it have rotational symmetry?

If the answer is affirmative, consider the following questions.

a What is the center of rotation?

b What is the order of symmetry?

c What conclusions can be drawn about the sides and angles of the parallelogram?

Answer

Yes, the parallelogram has rotational symmetry.

a The point of intersection of the diagonals.

2

c The opposite sides of a parallelogram are congruent as well as the opposite angles.

Hint

a How can the center of the parallelogram be found?

b Rotate the parallelogram

36 0^{\circ}

around its center and count the number of times it matches the original one.

c When the parallelogram is rotated

18 0^{\circ},

opposite vertices and sides are swapped. What does this mean in terms of the angle measures and side lengths?

Solution

To determine whether the parallelogram has rotational symmetry, it needs to be checked if a rotation of less than

36 0^{\circ}

about the center maps the parallelogram onto itself. In this case, the center of the parallelogram is the point of intersection of the diagonals.

As shown, a rotation of

18 0^{\circ}

about the intersection point of diagonals maps the parallelogram onto itself. Consequently, it is rotationally symmetric.

a The center of rotation is the point of intersection of the diagonals.

b Because rotations of

18 0^{\circ}

and

36 0^{\circ}

map the parallelogram onto itself, the order of symmetry is

2 .

c When

A B C D

is rotated

18 0^{\circ}

about its center, the following events occur.

The opposite vertices are swapped. That is, $A$ and $C$ are swapped as well as $B$ and $D .$
The opposite sides are swapped. That is, $\overline{A B}$ and $\overline{C D}$ are swapped as well as $\overline{B C}$ and $\overline{D A} .$

The first point implies that the measures of

∠ A

and

∠ C

are equal and the measures of

∠ B

and

∠ D

as well. Therefore, in a parallelogram, the opposite angles are congruent angles.

∠ A ≅ ∠ C ∠ B ≅ ∠ D

Similarly, the second point implies that the lengths of

\overline{A B}

and

\overline{C D}

are equal and the lengths of

\overline{B C}

and

\overline{D A}

as well. Therefore, in a parallelogram, the opposite sides are congruent segments.

\overline{A B} ≅ \overline{C D} \overline{B C} ≅ \overline{D A}

On the parallelogram, these congruences can be represented as follows.

Discussion

Properties of Parallelograms with Rotational Symmetry

The symmetries of a figure help determine the properties of that figure. For instance, since a parallelogram has $18 0^{\circ}$ rotational symmetry, its opposite sides and angles will match when rotated $18 0^{\circ},$ which allows for the establishment of the following property.

The opposite sides and angles of a parallelogram are congruent.

Symmetries can be used to characterize classes of figures. For example, all parallelograms have

18 0^{\circ}

rotational symmetry. The same is true for rectangles, rhombi, and squares.

Rotating 180 degrees a rectangle/rhombus/square

In consequence, it can be said that rectangles, rhombi, and squares are special kinds of parallelograms. Upon taking a closer glance, notice that all squares have

9 0^{\circ}

rotational symmetry, which implies squares belong to a particular class of parallelograms. Even more, this

9 0^{\circ}

rotational symmetry implies that all squares have congruent sides and angles.

Discussion

Defining Line Symmetry

Is there another type of symmetry apart from the rotational symmetry? The answer is yes. Some figures can be folded along a certain line in such a way that all the sides and angles will lay on top of each other. In this case, it is said that the figure has line symmetry.

Concept

Line Symmetry

A figure in the plane has line symmetry if the figure can be mapped onto itself by a reflection in a line. This line of reflection is called the line of symmetry. When a figure has line symmetry, it is said to be reflectionally symmetric or line symmetric.

An isosceles triangle with its line of symmetry

Some figures can have more than one line of symmetry.

Because of the symmetry definition, it seems logical to think that figures with congruent parts are symmetric. However, each figure needs to be studied separately before drawing conclusions.

Example

Line Symmetry in Parallelograms

After learning about line symmetry, Paulina and Heichi decided to study whether some quadrilaterals are line symmetric. To start, they picked a parallelogram.

Without thinking too much, Paulina said it has no line symmetry. However, Heichi said that the parallelogram is symmetric along the line connecting the midpoints of

\overline{A B}

and

\overline{C D} .

Who is correct?

Hint

Try to find a line along which the parallelogram can be bent so that all the sides and angles are on top of each other. The lines containing the diagonals or the lines connecting the midpoints of opposite sides are always good options to start.

Solution

To determine whether the parallelogram $A B C D$ is line symmetric, it needs to be checked if there is a line such that when $A B C D$ is reflected on it, the image lies on top of the preimage. Before start testing lines, mark the midpoints of each side.

In the event that

A B C D

is line symmetric, the lines connecting the midpoints of opposite sides are good candidates of being lines of symmetry. Also, the lines containing the diagonals could work. All possible lines can be tried in the following applet.

As can be seen, no reflection worked! In fact, there is no line along which the parallelogram can be reflected onto itself. This means that

A B C D

is not line symmetric. In conclusion, Paulina is right.

Example

Line Symmetry in Rectangles and Rhombi

Despite the previous example showing a parallelogram with no line symmetry, other types of parallelograms should be studied first before making a general conclusion. Consider a rectangle and a rhombus. Study whether or not they are line symmetric.

If both polygons are line symmetric, compare their lines of symmetry.

Answer

Both rectangles and rhombi have line symmetry. Rectangles have lines of symmetry connecting midpoints of opposite sides, while rhombi have lines of symmetry along its diagonals.

Hint

For each polygon, consider the lines along the diagonals and the lines connecting midpoints of opposite sides.

Solution

When studying a polygon's line symmetries, it is always good to consider the lines along the diagonals and the lines connecting midpoints of opposite sides. For simplicity, consider one polygon at a time, starting with the rectangle.

From the diagram, it can be seen that the rectangle is line symmetric. It has two lines of symmetries, each connecting the midpoints of opposite sides. Next, repeat the same steps with the rhombus.

As can be seen, the rhombus is line symmetric and also has two lines of symmetries. In contrast to the rectangle, however, the lines of symmetries are the lines containing its diagonals.

Discussion

Line Symmetry in Rectangles and Rhombi

Before, it was said that rectangles, rhombi, and squares are special kinds of parallelograms because they all have $18 0^{\circ}$ rotational symmetry. In addition, rectangles and rhombi have line symmetry.

Polygon	Line Symmetry
Rectangles	Along the lines connecting midpoints of opposite sides
Rhombi	Along the lines containing the diagonals

Consequently, based on the line symmetries, it can be concluded that rectangles and rhombi are not directly related. Therefore, they belong to different parallelogram classes. The next natural question is, what about the squares?

Example

Line Symmetry in Squares

Does a square have line symmetry?

If so, how many different lines of symmetry does it have?

Answer

Yes, squares are line symmetric. They have four lines of symmetries — two of them connect the midpoints of opposite sides and the other two contain its diagonals.

Hint

The lines containing the diagonals or the lines connecting the midpoints of opposite sides are always good options.

Solution

To determine whether the square is line symmetric, it is always good to consider the lines along the diagonals and the lines connecting midpoints of opposite sides. All possible lines can be tried in the following applet. Give it a go!

As might be verified, the square is indeed line symmetric. It has four lines of symmetry — two connect the midpoints of opposite sides and two others contain its diagonals.

Discussion

Summarizing Line Symmetry in Parallelograms

The line symmetries of a square confirm the claim made previously that squares belong to a particular class of parallelograms.

Polygon	Number of Line Symmetries	Line Symmetry
Rectangles	$2$	Along the lines connecting midpoints of opposite sides
Rhombi	$2$	Along the lines containing the diagonals
Squares	$4$	Two along the lines connecting midpoints of opposite sides and two along the lines containing the diagonals

Notice that two symmetries of the square correspond to the rectangle's symmetries and the other two correspond to the rhombus symmetries. This suggests that squares are a particular case of rectangles and rhombi.

Example

Line Symmetry in a Trapezoid

Paulina and Heichi have the task of investigating whether trapezoids have some symmetry. The teacher gave them the diagram below, but she told them to try any trapezoid they could imagine.

What conclusion should Paulina and Heichi reach?

Answer

Trapezoids have no rotational symmetry and have line symmetry only when the trapezoid is isosceles. In this case, the line of symmetry is the line passing through the midpoints of the bases.

Hint

A trapezoid is a quadrilateral with exactly one pair of parallel sides. Then, parallelograms are not trapezoids. Do not forget to try with isosceles trapezoids.

Solution

Consider any trapezoid

J K L M

where

\overline{J K}

and

\overline{L M}

are parallel. First, start investigating whether

J K L M

has rotational symmetry. To figure it out, set different dimensions for the trapezoid using the following applet. But remember, parallelograms are not trapezoids.

As investigated, regardless of the dimensions, no trapezoid has rotational symmetry. Next, repeat the same investigation but now checking for line symmetry.

From this last investigation, it may be seen that some trapezoids have line symmetry. Going a bit deeper, it can be concluded that these trapezoids are isosceles trapezoids.

A trapezoid has line symmetry only when it is isosceles trapezoid.

In this case, the line of symmetry is the line passing through the midpoints of each base.

Closure

Symmetry in Three-Dimensional Figures

Symmetries are not defined only for two-dimensional figures. The definition can also be extended to three-dimensional figures. In the real world, there are plenty of three-dimensional figures that have some symmetry. For example, sunflowers are rotationally symmetric while butterflies are line symmetric.

External credits: Alejandro Santillana

A three-dimensional figure is said to have axis symmetry if it can be rotated about a line onto itself by a rotation between

0^{\circ}

and

36 0^{\circ} .

This is the equivalent of rotational symmetry.

Hexagonal Prism with slider to rotate it

A three-dimensional figure has plane symmetry if there is a plane that divides the figure into two halves and each half is a reflection of the other across the plane. This is the equivalent of line symmetry.

A vertical triangular prism that is bisected by a perpendicular plane labeled P.

Level 1

Level 2

Level 3

Consider the following angles of rotation.

A 6 0^{\circ} C 9 0^{\circ} E 14 4^{\circ} B 7 2^{\circ} D 12 0^{\circ} F 18 0^{\circ}

Which of these options map the polygon onto itself? Select all that apply.

Examining the diagram, we see that this is an equilateral triangle. This means it has three congruent sides. If we draw a segment from each vertex to the center of the triangle, we create three congruent triangles.

Since the three triangles are congruent, the angles in the center have the same measure. Together, they sum to 360^(∘) which means each of them must have a measure that is one third of this. 360^(∘)/3=120^(∘) Therefore, if we rotate the triangle by 120^(∘), it will map onto itself.

We can also rotate the triangle by 240^(∘) and make it map onto itself. However, this is not represented by the options. The only viable option is D.

This quadrilateral has four congruent sides and four congruent angles. Therefore, this is a square. If we draw segments from each vertex to the center of the square, we create four congruent triangles.

Because the four triangles are congruent, the angles in the center have the same measure. Since they sum to 360^(∘), each of them must be a quarter of this. 360^(∘)/4=90^(∘) This means if we rotate the square by 90^(∘), it will map onto itself.

Also, we can also rotate it by 180^(∘) and make it map onto itself.

Therefore, both C and F are correct.

What is the least number of degrees the figure can be rotated by to make it map onto itself?

By drawing segments between the horizontal arrowheads and between the vertical arrowheads, we can see that if we rotate the figure by 90^(∘), it will map onto itself.

Therefore, this figure has a rotational symmetry of 90^(∘).

As in Part A, we will connect each of the star's five edges at the center of the star. This creates 5 congruent angles. Since a full turn is 360^(∘), we can calculate the measure of each congruent angle by dividing 360^(∘) by 5. 360^(∘)/5=72^(∘) Therefore, a 72^(∘) rotation about the center maps the star onto itself.

Can a figure have

9 0^{\circ}

rotational symmetry but not

18 0^{\circ}

rotational symmetry? Please explain your reasoning.

Can a figure have

18 0^{\circ}

rotational symmetry but not

9 0^{\circ}

rotational symmetry? Please explain your reasoning.

Let's consider a square because it is a shape with 90^(∘) rotational symmetry. Rotating a square 90^(∘) about its center maps it onto itself.

Rotating it 90^(∘) once more, it maps onto itself again.

Keep in mind that rotating a figure twice by 90^(∘) is the same thing as rotating it by 180^(∘). Therefore, it is impossible for a figure to have 90^(∘) rotational symmetry but not 180^(∘).

Let's consider a rectangle because it has 180^(∘) rotational symmetry. A shape with rotational symmetry of 180^(∘), can be flipped upside down and map onto itself.

Suppose the rectangle has 90^(∘) rotational symmetry, then we would be able to rotate it a quarter of a full turn and have it map onto itself. Try rotating it 90^(∘).

We see that this is not possible because one pair of sides is longer than the other. Therefore, it is possible for a figure to have 180^(∘) rotational symmetry but not 90^(∘).

How many lines of symmetry can you find in the following shape?

If we can fold a shape along a line such that the two parts map onto each other, we have found a line of symmetry. For this trapezoid, there is only one way we can fold it that fits this criterion.

As for the cross, we can fold it both horizontally and vertically and have it map onto itself.

However, we can also fold the cross along a 45^(∘) or a - 45^(∘) line and make the figure map onto itself.

For this parallelogram, there is no way of folding it such that the two parts map onto each other. Let's try folding it across a vertical line and see what happens.

Let's also try to fold the parallelogram across a horizontal line.

We could try other lines as well, but the parallelogram cannot be mapped onto itself no matter how we fold it. Therefore, it has no line of symmetry.

One line of symmetry we can see is a vertical line through the middle of the star.

However, we can also draw a line through each of the other four points of the star and fold it such that the star maps onto itself.

We have a total of 5 lines of symmetry.

How many lines of symmetry can you find in the following word?

M U M

L O O N

O X

For any word written using the English alphabet, we have two potential lines of symmetry. Either we fold the top half over the bottom half, or we fold the left half over the right half. Let's try drawing a vertical line of symmetry.

If we fold the left half over the right half, the left part of the U and the M on the left-hand side, will perfectly cover the right part of the U and the M on the right-hand side. Therefore, the word MUM has a vertical line of symmetry. Now let's try to fold the word across a horizontal line of symmetry.

As we can see, the top half does not match the bottom half. Therefore, the word does not have a horizontal line of symmetry. In conclusion, it only has one line of symmetry.

As in Part A, we will try to fold the word along a vertical line of symmetry.

As we can see, the O on the left-hand side and the O right-hand side will map onto each other when the word is folded along the vertical line of symmetry. However, the L and N do not map onto each other. Let's try folding the word across a horizontal line of symmetry.

Again, both O letters will map onto themselves when we fold the top half over the bottom half. However, the upper and lower half of L and N do not map onto each other. Therefore, the word LOON does not have any lines of symmetry.

Let's try folding the word OX horizontally and vertically.

The word OX has one line of symmetry.

Symmetries of Plane Figures

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