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### Course: Geometry (all content) > Unit 10

Lesson 8: 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?(121 votes)- The line of symmetry is the imaginary line where you could fold the image and have both halves match exactly.(215 votes)

- A circle has infinite lines of symmetry right?(45 votes)
- Yes.

Any straight line that passes through the center of a circle is also a line of symmetry of that circle.(45 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.(32 votes)

- Does the number of sides of the shape change the amount of lines of symmetry?(11 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.(11 votes)

- can rotational symmetry be used in organic shapes?(7 votes)
- yes, organic shapes such as snow flakes can have rotational symmetry.(11 votes)

- can there be a infinite number of lines of symmetry?(9 votes)
- Only circles because you can put the line of symmetry anywhere on a circle and still be symmetrical(0 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?(11 votes)
- Why the line of symmetry in rectangle have only two why not four(6 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!(9 votes)

- what is 721,390 x 635,495(8 votes)
- *I am not sure. What is it?*(2 votes)

- At0:16isn't the polygon actually a heptagon because it has seven sides?(4 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.