symmetry problem solving questions

Symmetry Questions

Symmetry Questions are a set of questions based on the topic of symmetry. These questions help the student in understanding the topic better. It also improves the spatial sense of the student and improves the thinking ability which is due to the imagination needed to solve the problems. Before answering the questions, for detailed information, visit symmetry .

For Synopsis: Symmetry is a wide topic that is specific to shapes and their appearances. Symmetry is a geometrical topic, where a shape is observed to have an imaginary line called a line of axis. This line of axis divides the shape in such a way that both the left side and right side of the line of axis are a mirror images to each other. Let us consider a simple example of the letter M. Note that M is also a shape obtained by straight lines. Now M can be split in a way such that both halves look like the mirror image of one another.

Topics to refer before answering Symmetry Questions are

• Lines of Symmetry :

The above example has only one line of symmetry. There are shapes which can have multiple lines of symmetry. Ex. Circle can have an infinite number of symmetrical lines.

• Rotation Symmetry.

When a shape is moved or rotated, if the shape appears the same, then the shape has a rotational symmetry

• Reflection Symmetry

Symmetry which has a mirror reflection. Same as letter M.

• Translational Symmetry

When an image is moved/displaced from its original position, but if the orientation remains the same.

• Glide Symmetry

When an image undergoes both Translational and reflection symmetry then we call it as glide symmetry

Let us start solving the Symmetry Questions.

Symmetry Questions With Solutions

Here are a few Symmetry questions with the answers.

Question 1:

How many English alphabets are symmetrical?

14 letters have lines of symmetry either horizontally or vertically.

They are A, B, C, D, E, H, I, K, M, O, T, V, W, X.

Question 2:

How many lines of symmetry does these geometrical shapes have?

• Square : has 4 lines of symmetry
• Pentagon: has 5 lines of symmetry
• Rhombus: has 2 lines of symmetry
• Trapezium: has 0 lines of symmetry

Question 3:

Which of the following shapes have rotational symmetry?

• A square has a rotational symmetry of order 4
• Circle has a rotational symmetry at every angle of the circle
• Hexagon has rotational symmetry of order 6
• Crescent has rotational symmetry of order 2

Question 4:

Draw the lines of Symmetry for the following shapes.

The Line of Symmetry for the given shapes are:

Question 5:

Show translation symmetry to the following images

• Square has been translated from the 1st Quadrant to the 3rd Quadrant.
• Hexagon Translated from the 2nd Quadrant to 3rd Quadrant
• Triangle translated from 2nd Quadrant to 4th Quadrant.

Question 6:

How do reflection symmetry look for the following shapes?

The reflection for the Star, Moon and Sun appear the same as the original images

Question 7:

Draw a shape that does not have symmetry.

Question 8:

Does the image have a symmetry along x- axis?

The given image does not have symmetry along x- axis. However it is symmetrical diagonally.

Question 9:

How to make the given image symmetrical along the y-axis?

Step 1: Remove the cell in the 1st row, 3rd column

Step 2: Add the 5th cell in the 1st column in the place of the 2nd column fourth row.

Question 10:

Colour the appropriate cells to make it symmetrical for the given image.

There are multiple ways to make the given image symmetrical. 3 methods are given below.

Colour all cells with the same colour. (either grey or white) to make it symmetrical

Video Lesson on Symmetry

Related Articles:

• Line of Symmetry
• Figures with Symmetry
• Reflection and Symmetry
• Rotational Symmetry

Practice Questions on Symmetry

• Draw a geometrical figure with no Symmetry.
• What is the rotational symmetry of a nonagon?

• What is the reflection symmetry for the letter M.

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Symmetry Worksheets Line Symmetry Easier

Welcome to the Math Salamanders Line Symmetry Worksheets page. Here you will find a range of free printable symmetry worksheets, which will help your child to practice their reflecting and flipping skills.

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Symmetry Help

The Math Salamanders have a large bank of free printable symmetry worksheets. Each symmetry sheet comes complete with answers for support.

Handy Hints

Each point or block that has been reflected must remain the same distance from the mirror line as the original point. So if point A is 3 squares away from the mirror line, then the reflection of point A must also be 3 squares away.

When reflecting a shape, look at the corners of the shape and reflect each corner first as a dot in the mirror line. The dots can then be joined up (in the correct order!)

For lines of symmetry at angles of 45°, it is often better to rotate your paper so that the line of symmetry is vertical or horizontal, and the rest of the paper is at an angle.

The basis and understanding of symmetry starts at about Grade 2, and then develops further in Grades 3,4 and 5.

Line Symmetry Worksheets

On this webpage you will find our range of line symmetry sheets for kids.

The sheets have been carefully graded with the easier sheets coming first. The first 3 worksheets involve only horizontal and vertical lines only. The next 3 worksheets involve reflecting diagonal lines as well.

There are also some templates at the end of this section for you to create your own shapes for your child to reflect, or, even better, for your child to create their own symmetric patterns!

Using these sheets will help your child to:

• learn to reflect a shape in a vertical or horizontal mirror line;
• learn to reflect a shape in both a vertical and horizontal mirror line;

Reflecting in 1 mirror line

Horizontal and vertical lines only.

• Line Symmetry 1
• PDF version
• Line Symmetry 2
• Line Symmetry 3
• Line Symmetry 4
• Line Symmetry 5
• Line Symmetry 6

Reflecting in Diagonal mirror lines

• Line Symmetry 9
• Line Symmetry 10

Reflecting in vertical, horizontal and diagonal lines

• Line Symmetry 11

Reflecting in 2 mirror lines

• Line Symmetry 7
• Sheet 7 Answers
• Line Symmetry 8
• Sheet 8 Answers

Reflecting in vertical and horizontal or 2 diagonal lines

• Line Symmetry 12

Line Symmetry templates

• Line Symmetry Template 1
• Line Symmetry Template 2
• Line Symmetry Template 3
• Line Symmetry Template 4
• Line Symmetry Template 5
• Line Symmetry Template 6

Looking for something harder?

Here you will find a range of line symmetry activity sheets with one or two mirror lines.

The sheets in this section are similar to those on this page, but are more complicated and at a harder level.

• Harder Symmetry Activities

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

Block Symmetry Worksheets

Here you will find a range of symmetry worksheets reflecting blocks instead of lines.

These sheets are at an easier level than the ones on this page.

• Symmetry Worksheets - Block Symmetry
• Explore 2d Shapes Worksheets

Looking for some geometry worksheets to get children thinking and reasoning about 2d shapes?

The shapes on this page are all about children really understanding what 2d shapes are all about, and using their reasoning skills to justify their thinking.

• know the properties of a range of 2d shapes;
• recognise that some shapes can also be described as being other shapes; e.g. a square is also a rhombus;
• recognise and understand right angles, parallel lines, lines of symmetry;
• develop their geometric reasoning skills.

Coordinate Sheets

Here is our collection of printable coordinate plane grids and coordinate worksheets.

Using these fun coordinate sheets is a great way to learn math in an enjoyable way.

• plot and write coordinates.
• Coordinate Plane Grid templates
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Captain Recommends

Have a look at these online symmetry games - a great way to learn symmetry and get instant feedback!

• Softschools Symmetry Game
• Sheppard Software Symmetry Activities

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Course: 4th grade   >   Unit 11

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Symmetry Worksheets

Symmetry worksheets give an ample amount of practice questions which could include drawing lines of symmetry , determining if a picture is symmetric and can also include reflection and flipping questions.

Benefits of Symmetry Worksheets

Symmetry is a concept that can be used in a variety of real life questions; hence, it is always better for students to have an understanding of this concept. By solving several practice problems on symmetry, students can master the subject quickly. Symmetry worksheets help students with this goal.

They provide questions in an increasing level of difficulty enabling students to comprehend concepts and build strong foundations. Due to the flexibility provided by the worksheets, it also gives them a fun learning experience. Students can also improve their analytical skills by solving the problems in these worksheets.

Download Symmetry Worksheet PDFs

These math worksheets should be practiced regularly and are free to download in PDF formats.

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Resources tagged with: Symmetry

There are 71 NRICH Mathematical resources connected to Symmetry , you may find related items under Transformations and constructions .

Always, Sometimes or Never? Shape

Are these statements always true, sometimes true or never true?

Use the information on these cards to draw the shape that is being described.

National Flags

This problem explores the shapes and symmetries in some national flags.

Poly Plug Pattern

Create a pattern on the small grid. How could you extend your pattern on the larger grid?

Exploded Squares

This practical activity challenges you to create symmetrical designs by cutting a square into strips.

Attractive Rotations

Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...

Counters in the Middle

This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

Reflector ! Rotcelfer

Can you place the blocks so that you see the reflection in the picture?

Coordinate Challenge

Use the clues about the symmetrical properties of these letters to place them on the grid.

Stringy Quads

This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!

Octa-flower

Join some regular octahedra, face touching face and one vertex of each meeting at a point. How many octahedra can you fit around this point?

Symmetry Challenge

How many symmetric designs can you make on this grid? Can you find them all?

Shady Symmetry

How many different symmetrical shapes can you make by shading triangles or squares?

Reflecting Squarely

In how many ways can you fit all three pieces together to make shapes with line symmetry?

A Cartesian Puzzle

Find the missing coordinates which will form these eight quadrilaterals. These coordinates themselves will then form a shape with rotational and line symmetry.

Attractive Tablecloths

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes".

Witch of Agnesi

Sketch the members of the family of graphs given by y = a^3/(x^2+a^2) for a=1, 2 and 3.

Folium of Descartes

Investigate the family of graphs given by the equation x^3+y^3=3axy for different values of the constant a.

Logosquares

Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares.

Mean Geometrically

A and B are two points on a circle centre O. Tangents at A and B cut at C. CO cuts the circle at D. What is the relationship between areas of ADBO, ABO and ACBO?

Colouring Triangles

Explore ways of colouring this set of triangles. Can you make symmetrical patterns?

Symmetric Trace

Points off a rolling wheel make traces. What makes those traces have symmetry?

Building Patterns

Can you deduce the pattern that has been used to lay out these bottle tops?

Emmy Noether

Find out about Emmy Noether, whose ideas linked physics and algebra, and whom Einstein described as a 'creative mathematical genius'.

Watch Those Wheels

Have you ever noticed the patterns in car wheel trims? These questions will make you look at car wheels in a different way!

What mathematical words can be used to describe this floor covering? How many different shapes can you see inside this photograph?

Plot the graph of x^y = y^x in the first quadrant and explain its properties.

An irregular tetrahedron has two opposite sides the same length a and the line joining their midpoints is perpendicular to these two edges and is of length b. What is the volume of the tetrahedron?

Symmetrical Semaphore

Someone at the top of a hill sends a message in semaphore to a friend in the valley. A person in the valley behind also sees the same message. What is it?

Hidden Meaning

What is the missing symbol? Can you decode this in a similar way?

Pattern Power

Mathematics is the study of patterns. Studying pattern is an opportunity to observe, hypothesise, experiment, discover and create.

Two Triangles in a Square

Given that ABCD is a square, M is the mid point of AD and CP is perpendicular to MB with P on MB, prove DP = DC.

Some local pupils lost a geometric opportunity recently as they surveyed the cars in the car park. Did you know that car tyres, and the wheels that they on, are a rich source of geometry?

Tournament Scheduling

Scheduling games is a little more challenging than one might desire. Here are some tournament formats that sport schedulers use.

Classifying Solids Using Angle Deficiency

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry

The Frieze Tree

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

Frieze Patterns in Cast Iron

A gallery of beautiful photos of cast ironwork friezes in Australia with a mathematical discussion of the classification of frieze patterns.

A red square and a blue square overlap. Is the area of the overlap always the same?

Square Pizza

Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

Prime Magic

Place the numbers 1, 2, 3,..., 9 one on each square of a 3 by 3 grid so that all the rows and columns add up to a prime number. How many different solutions can you find?

Rhombicubocts

Each of these solids is made up with 3 squares and a triangle around each vertex. Each has a total of 18 square faces and 8 faces that are equilateral triangles. How many faces, edges and vertices does each solid have?

An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?

The ten arcs forming the edges of the "holly leaf" are all arcs of circles of radius 1 cm. Find the length of the perimeter of the holly leaf and the area of its surface.

Eight Dominoes

Using the 8 dominoes make a square where each of the columns and rows adds up to 8

A Problem of Time

Consider a watch face which has identical hands and identical marks for the hours. It is opposite to a mirror. When is the time as read direct and in the mirror exactly the same between 6 and 7?

Maltese Cross

Sketch the graph of $xy(x^2 - y^2) = x^2 + y^2$ consisting of four curves and a single point at the origin. Convert to polar form. Describe the symmetries of the graph.

Sketch the graphs for this implicitly defined family of functions.

Find the shape and symmetries of the two pieces of this cut cube.

This activity investigates how you might make squares and pentominoes from Polydron.

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Symmetry is an easy and important concept which can be introduced to grade 4 kids. Symmetry worksheets here make kids understand that a line dividing an object or image into two equal halves, is known as line of symmetry and these split shapes are called symmetrical shapes. The split halves are mirror images.

Figures which cannot be divided into two equal halves (same size and shape) or which does not have a line of symmetry are asymmetrical figures.

Do all objects have only one line of symmetry? Consider different  2 dimensional  and 3 dimensional shapes and try drawing their line or lines of symmetry.

http://mathfunworksheets.com/2d-shapes/

http://mathfunworksheets.com/3d-shapes/

The answer is no. Not all shapes have only one line of symmetry. There are few objects which have two lines of symmetry. These objects can be divided into two equal halves in two ways.

More than two lines   of symmetry

Some figures have more than two lines of symmetry, which means they can be split into equal size and shape in more than two ways. The below shown examples are figures which exhibit three and more lines of symmetry.

Worksheets on  Reflection symmetry  are included here. Once familiar with symmetry, reflection symmetry is easy to deal with. These printable worksheets have practice pages to identify the line of symmetry, symmetrical and asymmetrical objects, counting the number of lines of symmetry, drawing the other half or the mirror image of the symmetrical object, cut- paste activity and much more. Download all and practice to the fullest.

Symmetry worksheets free download pdf

RELATED GEOMETRY TOPICS   :   2D Shapes ,  3D Shapes ,  Lines rays line segments and planes ,  Composing and Decomposing shapes

Symmetrical or Asymmetrical

Draw lines of symmetry

Number of lines of symmetry

Lines of symmetry - Real life objects

Complete the image

Complete the image - Grids

Draw the other half - Graph

Match the following

Cut Paste activity

Find the other half

Mirror Reflection

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Copyright © 2024. Math Fun Worksheets. All reserved!

• 1 Hidden symmetry
• 2 Symmetry with respect angle bisectors
• 3 Symmetry with respect angle bisectors 1
• 4 Construction of triangle
• 5 Symmetry with respect angle bisectors 2
• 6 Symmetry of radical axes
• 7 Composition of symmetries
• 8 Composition of symmetries 1
• 9 Composition of symmetries 2
• 10 Symmetry and secant
• 11 Symmetry and incircle
• 12 Symmetry and incircle A
• 13 Symmetry for 60 degrees angle
• 14 See also

Hidden symmetry

Symmetry with respect angle bisectors

Symmetry with respect angle bisectors 1

[email protected], vvsss

Construction of triangle

Construction

Symmetry with respect angle bisectors 2

Symmetry of radical axes

Composition of symmetries

It is known that the composition of two axial symmetries with non-parallel axes is a rotation centered at point of intersection of the axes at twice the angle from the axis of the first symmetry to the axis of the second symmetry.

Composition of symmetries 1

Composition of symmetries 2

Symmetry and secant

Symmetry and incircle

Symmetry and incircle A

Symmetry for 60 degrees angle

a) One can find successively angles (see diagram).

• Symmedian and Antiparallel

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• symmetry-quiz

Symmetry Quiz

Symmetry - practice problems with solutions.

7 Superb Symmetry Activities (with a FREEBIE!)

• Freebies , Math , Planning

In the hustle and bustle of the upper elementary classroom, where every moment is precious and every standard is vital, it can be difficult to give every last educational concept the attention it deserves. This is especially true when a concept only has one or two standards addressing it. If it only has one standard in all of upper elementary, then it must not be too important, right?

… Or, maybe, the concept is just SO important that it only needs to be covered once to make a lasting impact on your kiddos! That’s what I’m choosing to believe, anyway, when it comes to symmetry. Maybe it’s my artistic side speaking, but I absolutely LOVE symmetry, and it actually hurts my feelings that this geometry concept doesn’t always get the attention it deserves.

For instance, when conducting my research for this blog post, I noticed a severe lack of educational resources and ideas that align with the Common Core State Standard 4.G.3 : “Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.”

It’s obviously important to ensure your kiddos are meeting CCSS, but once that goal is met, symmetry can sometimes fall to the wayside. It’s understandable, since there are simply SO MANY standards to get to, but symmetry really is important in its own right!

Let’s check out why it’s vital to students’ mathematical careers, then we’ll dive into seven symmetry activities that are sure to supplement your geometry lessons… Complete with a FREEBIE at the end!

Why Bother with Symmetry?

It’s not just important to cover symmetry in preparation for testing. Exploring symmetry helps students build fluency in math by reinforcing key concepts like congruence, transformations, and geometric properties. By working through engaging, hands-on symmetry activities and discussions, students develop a stronger understanding of math principles and develop fluency in applying them to new situations.

In addition, building a strong foundation in symmetry also helps students develop spatial reasoning skills, which are super important for understanding tougher geometric concepts and spatial relationships down the line. It’s a great way to help students learn to visualize shapes, identify patterns, and make predictions.

There’s also plenty of symmetry to be found in nature, too! This is certainly a neat way to make real-world connections with your kiddos, but beyond that, it also provides you with a great opportunity for an integrated curriculum with Earth science… Which, as you know, is like, my favorite thing ever . 😉

Okay, I think I’ve made my point. Let’s jump right into the activities!

Seven Symmetry Activities

1. symmetry art swap.

For this activity, you’ll want to provide individuals with graph paper and art supplies. Every student will begin by creating a “line of symmetry,” preferably horizontal or vertical to begin with, through the center of the page. Then, they can create a geometric design of their liking on ONE side of the mirror line. Of course, students should be given free agency with their designs, but you may want to encourage simpler designs to start with to allow kiddos to get the hang of the concept of symmetry.

Once students have created their designs on one side of the mirror line, have them swap their graph paper with a partner. The partner should then try to finish the design on the other side of the symmetry line! Since this is a pretty quick activity, I recommend having students repeat these steps a few more times. They can experiment with more complex designs and even add colors to be mirrored!

It’s not a bad idea either to encourage kiddos to find NEW partners every time, too. You don’t want them sticking to only their best friend—switch it up so everyone gets a diverse experience.

These symmetry art pieces can be shown off in the classroom or even a hallway bulletin board for a cute, easy display. Kiddos love showing off their work, and a splash of color in the halls always does wonders to brighten spirits!

2. Symmetry Manipulatives

Similar to the art swap above, this is another way for students to express their creativity by creating symmetrical designs. This time, though, they can only use manipulatives, such as pattern blocks, tangrams, or other physical geometrical shapes. It’s a bit more challenging than the art swap, since they don’t have complete freedom over the design; rather, they’re bound by the dimensions of the trapezoids and rhombuses and squares!

For this activity, have students sit across from one another and try to replicate one another’s designs using the pattern blocks. They should notice that the design is “mirrored” across a horizontal line of symmetry between them. Kiddos can even take turns making designs and creating a symmetrical version.

3. Symmetry Construction

If someone had told me back at university just how much I’d end up advocating for the use of toothpicks in the classroom, I would never have believed them… But, goodness, there’s no denying their myriad of uses! From fencing in animals to investigating camouflage , toothpicks are practically the backbone of hands-on activities.

And, as you’re sure to see, geometry is no exception! For this activity, have students use toothpicks to create symmetrical shapes. They can connect the toothpicks (or even popsicle sticks, if you prefer) at their endpoints to form geometric shapes, then identify and label the lines of symmetry within their creations—or you can have a partner do that part.

Challenge students to see how many lines of symmetry they can create within one design! Of course, in that case, you may want to ban circles… That might be a little too easy. 😉

4. Symmetry Scavenger Hunt

Listen, I know I recommend scavenger hunts, like, ALL the time, but you can’t blame me: they’re a perfect way to get students up and moving around, keeping them engaged while staying on-task!

For this activity, you’ll encourage kiddos to engage with a geometric scavenger hunt around the classroom, encouraging them to search for symmetrical objects or patterns. If you’d like, you can set out symmetrical pictures or the like ahead of time, but that’s not totally necessary; most classrooms are already chock-full of symmetry if you take the time to look!

Have students record their findings in a notebook or on a piece of paper as they move through the scavenger hunt. They should sketch all of the items they discover that are symmetrical and draw its lines of symmetry—for instance, they could draw the top-down view of a desk and include all of its symmetrical lines.

Once students have had enough time to scan the entire room, reconvene as a class and discuss their findings. Encourage productive discussion by asking questions, such as “Which objects have the most symmetrical lines?” and even deeper critical-thinking questions like “Would the entire classroom be symmetrical if it were mirrored? Why or why not?”

5. Symmetry Nature Walk

What’s this? Another opportunity to integrate curriculum? You got it! Add a helping of Earth science to your geometry unit by bringing students outside to make mathematical observations about nature.

Before heading outside, however, it’s time for a brainstorm sesh. As a class, discuss what kinds of symmetrical objects you may find outside! This is a good way to get students visualizing symmetry, which is a vital step to fully grasping geometry.

Then… Go outside! Just like with the scavenger hunt, have kiddos bring along paper in order to sketch images of the symmetrical objects they find along with their lines of symmetry. Some common examples they may find outside are leaves, flowers, and even insects!

After the adventure, reconvene in the classroom and have students share their findings. Ask more questions, like which objects outside had the most lines of symmetry, if there were any perfectly symmetrical objects (like circles) and so on.

This is a great opportunity for critical thinking, too! What observations can your students make about symmetry in nature? Is it common? Lots of wildlife is symmetrical to some degree, like humans. Why do they think this is? There are no right or wrong answers; the exercise is simply meant to get students thinking and visualizing!

6. Symmetry Stations

Next up, set up several different stations around the room that each center around a type of geometric shape, such as circles and ovals, triangles, parallelograms, and other shapes, like hearts, crosses, arrows, stars, and so on.

Each station should have around five “cards” each that have a different shape relevant to the station’s theme. For example, the triangle station should feature five different triangles of different sizes, each with varying amounts of lines of symmetry.

At each station, students should sketch the shapes on the cards on their own personal graph paper. Then, they can fold their paper to find lines of symmetry! The folding is an important part—not just because it’s hands-on, but also because it ensures that students are fully grasping the concept, specifically as it’s cited in CCSS 4.G.3 : “Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts.”

Students should work together with groups or partners and go through all of the stations until they have “dissected” all of the cards at each center. Then, go over the answers as a class and discuss any questions or thoughts your kiddos may have.

7. Symmetry Assessment Activities

Now that your students have explored the nuances of symmetry, it’s time to check for understanding! Let’s start with a couple informal assessments .

Ticket-out-the-doors are a great method here, as you can simply draw half of a shape on the whiteboard and have students copy that shape onto a piece of scrap paper. Then, they would finish drawing its other side, taking care to keep the shape symmetrical. Once they turn in the paper, you can easily see right off the bat who has grasped the concept and who may still need a bit more help to properly visualize symmetry.

Another informal assessment to use here is Four Corners. Assign each corner of your classroom a number, one through four. This number will represent the lines of symmetry that a shape has. Then, draw a shape on the whiteboard and have students move to whichever corner of the room matches the number of lines of symmetry they think the shape has.

It’s definitely a plus to get students up and moving around, but if you’re pressed for time, no worries! Instead of using Four Corners, you can simply draw the shape on the whiteboard, then have your students raise up the number of fingers that match how many lines of symmetry the shape has.

If you’re finding your kiddos can still use a bit of practice, or if you just want to incorporate some review time, I’ve got you covered: check out this symmetry FREEBIE pulled straight from my fourth grade math workshop geometry unit! Click the pic below or the link above to download the worksheet. And, hey, if you like that, then you can find the full 4th Grade Math Workshop Geometry Unit here !

With this array of ideas to choose from, you and your kiddos are ready to grab your compass, sharpen your pencils, and take off on an adventure of symmetry! Don’t let this math concept just be an addendum to your geometry unit; give symmetry the attention it deserves in your upper elementary classroom.

And, as always, be on the lookout for more engaging, easy-to-implement ideas for your math and science students. While you’re here, sign up for my email list so you won’t miss out!

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Lines of Symmetry

Lines Of Symmetry

Here we will learn about lines of symmetry , including symmetry properties within polygons, angle properties, and symmetry of different line graphs.

There are also lines of symmetry worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What are lines of symmetry?

Lines of symmetry are straight lines that divide a shape into two equal parts where one part is an exact reflection of the other.

A rectangle has two lines of symmetry (LoS) shown below using a dashed line. In the diagram on the left, the vertical line of symmetry splits the rectangle into two equal shapes, in the diagram on the right the horizontal line of symmetry splits the rectangle into two equal shapes.

If you imagine folding these rectangles along the line of symmetry, both sides will match perfectly.

To draw a line of symmetry, you need to locate the position where you can split the shape into two equal parts. This can be described as a mirror image.

Lines of symmetry in regular shapes

Many 2D shapes have lines of symmetry.

Regular polygons all share the property that the number of sides is equal to the number of lines of symmetry.

Below is a couple of examples of these cases:

A reason why regular shapes have many lines of symmetry is due to the angles within the shape being the same.

Lines of symmetry in irregular shapes

As soon as the angles in two-dimensional shapes change from their equal property, the number of lines of symmetry changes.

Let’s look at different shapes (specifically quadrilaterals) and their number of lines of symmetry.

The parallelogram is an interesting case because although it has set angle properties (the opposing angles are equal), they do not contain any lines of symmetry . 

They do however have rotational symmetry (see the lesson on rotational symmetry for more information).

It is important to note that the different orientations of each quadrilateral above will still result in the same number of lines of symmetry.

For example, if the kite rotates 90 degrees clockwise about it’s centre, the line of symmetry will still be the longest diagonal as the line of symmetry moves with the shape. Also, lines of symmetry do not have to go through two vertices.

Lines of symmetry in graphs

We don’t stop at shapes when we look at lines of symmetry, we can also consider lines of symmetry with different types of graphs.

Below is the graph of the equation y=x^2-4:  

Here we can see that the y -axis is a line of symmetry because the graph would be symmetrical for each point of the curve. This is a really useful tool when drawing quadratic graphs because each point on the curve will be symmetrical with another. 

Furthermore, we can express the line of symmetry to be the line x=0 as this is the equation of the vertical line for all points when x=0.

For more information on equations of straight line graphs, see the lesson equation of a line .

A straight line also has a line of symmetry. Take the straight line y=x:

If we constructed a line of symmetry at the origin, the line would be perpendicular (at 90 degrees to) the original line. The equation of this line is y=-x.

As the line continues to infinity in each direction, the two halves of the bisected line are the same length.

For more information on perpendicular lines, see the lesson parallel and perpendicular lines (coming soon).

Lines of symmetry in circles

A regular hexagon has 6 lines of symmetry, a decagon has 10 lines of symmetry, and an icosagon has 20 lines of symmetry. But what about a circle? 

A circle can be rotated around its centre and the shape will remain identical as the radius is the same for every point.

The diameter of a circle splits the circle into two equal sized semicircles. The diameter can be placed anywhere in the circle and by its definition, it must go through the centre of the circle so we will always have two equal parts.

This mirror symmetry can therefore be viewed along any diameter and so the circle is said to have infinite lines of symmetry. 

How to draw lines of symmetry

In order to draw lines of symmetry:

• Locate the centre of the 2D shape.
• Use a ruler to visualise a horizontal and/or vertical line of symmetry through the centre of the shape.

Continue to rotate the ruler around 180 degrees over the centre point to cover all sides and vertices.

Explain how to draw lines of symmetry

Lines of symmetry worksheet

Get your free lines of symmetry worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Lines of symmetry examples

Example 1: the equilateral triangle.

Draw all of the lines of symmetry for the equilateral triangle below:

Locate the centre of the 2D shape .

Draw a small x in the centre of the triangle:

2 Use a ruler to visualise a horizontal and/or vertical line of symmetry through the centre of the shape .

Here we can draw a vertical line as this divides the shape into two identical right angle triangles (one is a reflection of the other):

3 Continue to rotate the ruler around 180 degrees over the centre point to cover all sides and vertices.

As the shape has an odd number of vertices, we need to pass through a vertex with the opposing side. We cannot pass through opposing vertices as they don’t exist for this shape.

The equilateral triangle has 3 lines of symmetry.

Example 2: the regular pentagon 

Draw all of the lines of symmetry for a regular pentagon:

Draw a small x in the centre of the pentagon (this does not have to be exact):

Use a ruler to visualise a horizontal and/or vertical line of symmetry through the centre of the shape .

Here we can draw a vertical line as this divides the shape into two identical shapes (one is a reflection of the other):

Continue to rotate the ruler around 180 degrees over the centre point to cover all sides and vertices .

As the shape has an odd number of vertices, we need to pass through a vertex with the opposing side.

The regular pentagon has 5 lines of symmetry.

Example 3: using angles

Show that the hexagon below has no lines of symmetry.

Draw a small x in the centre of the hexagon (this does not have to be exact):

Imagine drawing a line from A to D . If this is a line of symmetry then the angles B and F, and C and E must be equal to one another. As C and E are not equal to each other, the line AD is not a line of symmetry.

As the shape has an even number of vertices, we need to pass through opposite vertices and then pass through opposing sides. When we pass through opposing vertices, we get:

When we pass through opposing sides, we get:

As there is no line that can be drawn through opposing vertices or sides of the shape that splits the shape into two reflected parts, this shape has no lines of symmetry.

Example 4: angle facts

Use angle facts to find the number of lines of symmetry in the following shape ABCDEF :

To find the centre of the shape, join the diagonals together. Here we have:

Despite the centre being outside the shape, we can use it to locate lines of symmetry. Next we need to calculate all of the interior angles of the shape to see if they can help us spot any lines of symmetry.

Angle CDE =180-45=135^o (angles on a straight line total 180^o )

Angle CFE = Angle CFA = (360-90) \div 2 = 135^o (angles at a point total 360^o )

Angle DEF = Angle BAF = 180-135 = 45^o (supplementary to angle CDE )

Angle BCF = Angle DCF = 90\div2 = 45^o

We now have the following information:

By sketching a vertical and horizontal line through the centre, we can see that the shape does not have a line of symmetry in either of these directions.

By rotating the ruler around the centre, we can see that the line CF splits the shape ABCDEF in half. Looking at the angles on either side of the line of symmetry, we can see that the angles at B and D , and the angles at A and E are the same.

As the other two angles are bisected, the line CF is the only line of symmetry in this shape.

Example 5: quadratic graph

Write the equation of the line of symmetry for the quadratic equation y=x^2+6x+8. :

Here, rather than locating the centre, we are going to locate the turning point of the curve. The turning point is the point on the curve where the gradient changes from positive to negative, or viceversa.

For the graph of y=x^2+6x+8 , we can see that the turning point is at (-3,-1) however for questions without a visual representation, the turning point can be calculated by completing the square:

\begin{aligned} &y=(x+3)^2-9+8\\\\ &y=(x+3)^2-1 \end{aligned}

Turning point: (-3,-1) .

Here we can see that if we sketch a vertical line through the turning point (-3,-1) , the coordinates on either side of this line are the same distance away from the line.

This means that the vertical line that intersects the x axis at -3 is a line of symmetry, or x=-3.

By rotating the ruler around the turning point, there are no other lines that show a symmetry in the graph of y=x^2+6x+8 . The only line of symmetry therefore is x=-3 .

Example 6: reciprocal graph

State the equation of the lines of symmetry for the equation y=\frac{4}{x} .

The graph of y=\frac{4}{x} has two asymptotes. Asymptotes are lines that are reached when a value is undefined, in this case when x=0, or y=0 . This is because, as the value of x gets larger (in the positive or negative direction), the value of y gets closer to zero.

This is also true as the value of x tends towards zero as we will be dividing 4 by a very small number, leaving us with an infinitely large value of y. The importance of this fact allows us to locate the “centre of the graph”. For this graph, the centre is at (0,0) .

Let us talk about the graph using the four quadrants:

If you place your ruler on the y -axis, we can see that Quadrants 1 and 4 are not symmetrical to Quadrants 2 and 3 . If we do the same with the x -axis, Quadrants 1 and 2 are not symmetrical to Quadrants 3 and 4 . So this graph has no horizontal or vertical lines of symmetry.

Continue to rotate the ruler around 180 degrees over the centre point to cover all sides and vertices

As we rotate the line of symmetry around the centre point, we can see that when we reach an angle of 45 degrees, the two sides of the line are symmetrical. The equation of this line is y=x.

If we continue rotating, we get to 135 degrees. This is the line y=-x . This means that the graph of y=\frac{4}{x} has two lines of symmetry:

\begin{aligned} &y=x\\\\ &y=-x \end{aligned}

Common misconceptions

• Rotational symmetry

Lines of symmetry are mixed up with rotational symmetry. A line of symmetry divides the shape equally into two symmetrical pieces. A rotational symmetry is the number of times a shape fits into itself when rotated around its centre.

• The diagonals of quadrilaterals

The diagonals of a quadrilateral are often incorrectly joined together and highlighted as a line of symmetry. The only quadrilateral where this is true is the square. 

Below is the example for an incorrect line of symmetry for a rectangle.

• The number of sides = the order of  lines of symmetry

Although this is true for regular shapes, this is not true for all shapes.

Related lessons

Lines of symmetry is part of our series of lessons to support revision on symmetry. You may find it helpful to start with the main symmetry lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

Practice lines of symmetry questions

1. Calculate the number of lines of symmetry for the kite below.

2. Calculate the number of lines of symmetry for a hexagon.

3. How many lines of symmetry does the octagon below have?

4. Calculate the number of lines of symmetry for the shape ABCDEFG .

5. State the equation of the line of symmetry for the graph y=x^2+20x-90

6. The graph of y=-\frac{1}{x} has two lines of symmetry. One of these lines is y=x . State the equation of the other line of symmetry.

Lines of symmetry GCSE questions

1. ABCD is a quadrilateral. Lines AB and CD are parallel. By finding the value for x , show that the shape has one line of symmetry.

2x+10+66=180

2x=104 so x=52^o

2\times52+10=114^o and 52+14=66^o

As angle BAD = ABC and angle ADC = BCD, ABCD   is an isosceles trapezium with 1 vertical line of symmetry.

(a) Complete table of values for the quadratic y=x^2+4x-10. \begin{aligned} &x \quad -2 \quad -1 \quad\quad 0 \quad\quad 1 \quad\quad 2 \quad\quad 3 \\ &y \quad \quad \;\: \quad-3 \quad\quad 0 \quad\quad 1 \end{aligned}

(b) Use the table to state the equation of the line of symmetry for the equation y=x^2+4x-10.

The values of y are symmetrical around the point (1,1)

3. Calculate the equation of the line of symmetry for the line y=2x+3 at the point (2,7) .

Perpendicular gradient m=-0.5

y=-0.5x+c at (2,7) means 7=-0.5 \times 2+c, \;c=8

Learning checklist

You have now learned how to

• Identify lines of symmetry in 2D shapes presented in different orientations

The next lessons are

• Pythagoras’ theorem
• Enlargement

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• A triangle has sides A, B, and C. The angle between sides A and B is #(pi)/3# and the angle between sides B and C is #pi/6#. If side B has a length of 26, what is the area of the triangle?
• How do you write the following in trigonometric form and perform the operation given #(sqrt3+i)(1+i)#?
• A triangle has sides A, B, and C. The angle between sides A and B is #(2pi)/3#. If side C has a length of #32 # and the angle between sides B and C is #pi/12#, what is the length of side A?

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AI Math Solver Question Helper 4+

Ai tutor to help math question, damjibhai chotaliya, designed for iphone.

• Offers In-App Purchases

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Description.

Using the latest AI, AI Math Solver Question Helper helps streamline your studies with tools for scan, writing, problem solving, help to homework and more. Simply click photograph of your math problem, and AI Math Solver Question Helper will review it, offering in depth insights and step-by-step breakdowns. This approach to understanding solutions and enhances your math abilities. AI Math Solver Question Helper is an advanced tool designed to assist students at all learning levels. By simply taking a photo of a math problem, users get instant, comprehensive solutions with detailed step by step explanations. The app covers a wide range of topics, from basic arithmetic to complex calculus, transforming challenging problems into manageable tasks. Its user-friendly design and powerful AI technology promote a deeper understanding of mathematical concepts and enhance problem solving skills. Ideal for homework help, exam preparation, or mastering difficult concepts, AI Math Solver Question Helper makes learning math both efficient and engaging. FEATURES INCLUDED: - Scan to Solve with AI Problem Solver - Get quick Word problem & Geometry solution - Math, Science, & Liberal Arts any subject can explained - Step by Step Explanations for any Query - Customized AI Tutor COVERED MATH TOPICS: - Solve Math problems for elementary to college and beyond every level - Algebra 1 & 2: covering linear, quadratic, exponential equations, and more - Calculus includes limits, differential equations, integrals, and more - Trigonometry: angles, functions, identities, and more - Simplify, factor, and evaluate polynomials and fractions - Matrix operations: multiplication, arithmetic, and more - Probability and statistics: including the distributive property, and more - Arithmetic: basic operations, number theory, and calculations Get AI Math Solver Question Helper to revolutionize study, solve math's problem, and to get help for your Homework. PREMIUM PLANS AVAILABLE: Get paid subscription to discover more advance features. - No Ads Anymore - Duration wise package AI Math Solver Question Helper for 1 Week - 6.99 USD Your payment will be charged to your iTunes Account as soon as your purchase is confirmed. More Details: Privacy Policy: https://damjibhaichotaliya.blogspot.com/2024/08/damjibhai-chotaliya-privacy-policy.html Terms of Use: https://damjibhaichotaliya.blogspot.com/2024/08/damjibhai-chotaliya-terms-of-use.html

App Privacy

The developer, Damjibhai Chotaliya , indicated that the app’s privacy practices may include handling of data as described below. For more information, see the developer’s privacy policy .

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Information

English, Arabic, Dutch, French, German, Hindi, Indonesian, Italian, Japanese, Korean, Malay, Portuguese, Punjabi, Russian, Simplified Chinese, Spanish, Thai, Turkish, Vietnamese

• AI Math for 1 Week $6.99
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