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### Course: High school geometry > Unit 2

Lesson 4: Symmetry# Intro to reflective symmetry

Introduction to the concept of an "axis of symmetry". Created by Sal Khan.

## Want to join the conversation?

- What does
`line of symmetry`

mean and how can you apply it to a shape? because i don't understand that at all and what can you not use it on?(125 votes)- The line of symmetry is the imaginary line where you could fold the image and have both halves match exactly.(222 votes)

- A circle has infinite lines of symmetry right?(46 votes)
- Yes.

Any straight line that passes through the center of a circle is also a line of symmetry of that circle.(49 votes)

- i thought the line of symmetry was where you could fold the image and have both hales match exactly(22 votes)
- You thought right! Some shapes cannot have a line of symmetry, but for those, which can, an imaginary line (sometimes called the axis) runs through the center cutting the figure into two identical parts. Keep in mind, the two parts will often be mirror images, depending on the figure.(35 votes)

- Does the number of sides of the shape change the amount of lines of symmetry?(12 votes)
- It does change, but depends more on if the polygon is regular or not. If a polygon is irregular, chances are there will be less lines of symmetry. An equilateral triangle will have 3 lines of symmetry, but a scalene triangle will have none. If a polygon is regular, the number of sides it has is the amount of symmetry lines it will have.(12 votes)

- This concept makes absolutely no sense. I have tried to do it over and over, and I am having no luck. I reflect the point over the line as Sal says, and it moves onto the other point. But the hint says it doesn't. can anyone help me with this?(13 votes)
- I honestly doubt this will help... but lemme know! When you reflect a figure across a line, the original figure and its image share a line of symmetry, which we have called the line of reflection. When you reflect a figure across a line and then reflect the image across a line that intersects the first line, your final image is a rotation of the original figure. The center of rotation is the point at which the two lines of reflection intersect. The angle of rotation is determined by connecting the center of rotation to a pair of corresponding vertices on the original figure and the final image.(4 votes)

- does a circle have infinite lines of symmntry(7 votes)
- Yes, it has infinite numbers of lines of symmetry.(14 votes)

- can rotational symmetry be used in organic shapes?(7 votes)
- yes, organic shapes such as snow flakes can have rotational symmetry.(12 votes)

- can there be a infinite number of lines of symmetry?(9 votes)
- a circle has an infinite numbers of symmetry(3 votes)

- Why the line of symmetry in rectangle have only two why not four(5 votes)
- Great question!

A line of symmetry must not only divide the figure into two congruent parts, but also must be**perpendicular**to every segment that connects a point in one part to the corresponding point in the other part.

While a diagonal divides a rectangle into two congruent triangles, that alone doesn’t necessarily imply that it is a line of symmetry.

Let’s say we have a non-square rectangle ABCD (in counterclockwise order) with diagonal BD. In the two congruent triangles formed, point A must correspond to point C. However, the segment connecting A and C is**not perpendicular**to diagonal BD. So diagonal BD is not a line of symmetry. For a similar reason, diagonal AC is not a line of symmetry.

Therefore, the two diagonals of a non-square rectangle do not count as lines of symmetry!

Have a blessed, wonderful day!(13 votes)

- what is a symmetry line this dosent help(5 votes)
- Symmetry occurs in many areas of mathematics. This lesson explains symmetry in math and explores the three basic types of symmetry: rotational symmetry, reflection symmetry, and point symmetry.

Definition of Symmetry

Symmetry comes from a Greek word meaning 'to measure together' and is widely used in the study of geometry. Mathematically, symmetry means that one shape becomes exactly like another when you move it in some way: turn, flip or slide. For two objects to be symmetrical, they must be the same size and shape, with one object having a different orientation from the first. There can also be symmetry in one object, such as a face. If you draw a line of symmetry down the center of your face, you can see that the left side is a mirror image of the right side. Not all objects have symmetry; if an object is not symmetrical, it is called asymmetric.

When working with symmetry, the initial image is called the pre-image, and the second image is called the image because it is the final step in the process. Just like the answer to a math problem is the final step in that process, the image is what is created when you rotate something 90 degrees or flip it about the x-axis. There are three basic types of symmetry: rotational symmetry, reflection symmetry, and point symmetry.

Reflection Symmetry

Sometimes called line symmetry or mirror symmetry, reflection symmetry is when an object is reflected across a line, like looking in a mirror. The face from before shows you reflection symmetry. Here are some more examples of reflection symmetry. The line of symmetry does not have to be vertical; it can go in any direction. Also, certain objects, like a square or a circle, can have many lines of symmetry.(3 votes)

## Video transcript

For each of these
diagrams, I want to think about whether
this blue line represents an axis of symmetry. And the way we can tell is if
on both sides of the blue line we essentially
have mirror images. So let's imagine. Let's take this top part of
this polygon, the part that is above this blue line
here, and let's reflect it across the blue line--
you could almost imagine that it's a reflection
over some type of a lake or something-- and see if we
get exactly what we have below. Then this would be
an axis of symmetry. So this point right over here,
this distance to the blue line, let's go-- the same
amount on the other side would get you right there. And so you immediately
see we start ending up with a point that is off
what's actually here in black, the actual bottom
part of the polygon. So this is a pretty good
clue that this is not an axis of symmetry. But let's just continue it,
just to go through the exercise. So this point, if you reflected
it across this blue line, would get you here. This point-- I'll do it
in a different color. This point, if you
were to reflect it across this blue
line, it would get you-- let me make sure I can
do that relatively straight. I can do a straighter
job than that. So if you go about
that distance about it, and I want to go straight
down into the blue line, and I'm going to go the same
distance on the other side, it gets me to
right around there. And then this point
over here, if I were to drop it
straight down, then if I were to go the same
distance on the other side, it gets me right around there. And then finally, this point
gets me right around there. So its mirror image
of this top part would look something like this. My best attempt to draw
it would look something like this, which is very
different than the part of the polygon that's
actually on the other side of this blue line. So in this case, the blue line
is not an axis of symmetry. So this is no. No, this blue line is
not an axis of symmetry. Now let's look at it over here. And our eyes pick this
out very naturally. Here you can see that it looks
like this blue line really divides this polygon in half. It really does look
like mirror images. It really does look
like, if you imagine that this is some type
of a lake, a still lake, so I shouldn't
actually draw waves, but this is some type of a lake,
that this is the reflection. And we can even go
point by point here. So this point right over
here is the same distance from, if we dropped a
perpendicular to this point as this one right over
here; this one over here, same distance, same distance
as this point right over here; and we could do that
for all of the points. So in this case,
the blue line does represent an axis of symmetry.