Points on line of reflection | Transformations | Geometry | Khan Academy

Points on line of reflection

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- [Voiceover] We're asked to
use the "Reflection" tool to

define a reflection that
will map line segment ME,

line segment ME, onto the
other line segment below.

So we want to map ME to
this segment over here

and we want to use a Reflection.

Let's see what they expect from us

if we want to add a Reflection.

So if I click on this it says

Reflection over the line from,

and then we have two coordinate pairs.

So they want us to define
the line that we're going

to reflect over with
two points on that line.

So let's see if we can do that.

To do that I think I need
to write something down

so let me get my scratch pad out

and I copied and pasted the same diagram.

And the line of reflection,
one way to think about it,

we want to map point E,
we want to map point E,

to this point right over here.

We want to map point M

to this point over here.

And so between any point
and its corresponding point

on the image after the reflection,

these should be equidistant
from the line of reflection.

This and this should
be equidistant from the

line of reflection.

This and this should be E,
and this point should be

equidistant from the line of reflection.

Or another way of thinking about
it, that line of reflection

should contain the midpoint
between these two magenta points

and it should contain the
midpoint between these two

deep navy blue points.

So let's just calculate the midpoints.

So we could do that with a
little bit of mathematics.

The coordinates for E
right over here, that is,

let's see that is

x equals negative four,

y is equal to negative four,

and the coordinates for the
corresponding point to E

in the image.

This is x is equal to
two, x is equal to two,

and y is equal to negative six.

So what's the midpoint between

negative four, negative four,

and two, comma, negative six?

Well you just have to take
the average of the x's

and take the average of the y's.

Let me do that, actually
I'll do it over here.

So if I take the average
of the x's it's going to be

negative four, negative four,

plus two, plus two,

over two, that's the average of the x's.

And then the average of
the y's, it's going to be

negative four plus negative six over two.

Negative four plus negative six,

over two and then close the parentheses.

Let's see, negative four
plus two is negative two,

divided by two is negative one.

So it's going to be negative one, comma.

Negative four plus negative six,

that's the same thing as
negative four minus six

which is going to be negative 10.

Divided by two is negative five.

Let me do that in a blue color

so you see where it came from.

Is going to be negative five.

So there you have it.

That's going to be the midpoint between E

and the corresponding point on its image.

So let's see if I can plot that.

So this is going to be,

this point right over here is going to be

negative one, comma, negative five.

So x is negative one,

y is negative five.

So it's this point right over here

and it does indeed look like the midpoint.

It looks like it's
equidistant between and E

and this point right over here.

And so this should sit on
the line of reflection.

So now let's find the midpoint
between M and this point

right over here.

The coordinates of M
are x is negative five,

and y is equal to three.

The coordinates here
are x is equal to seven

and y is equal to negative one.

So the midpoint, the x
coordinate of the midpoint,

is going to be the
average of the x's here.

So let's see it's going
to be negative five

plus seven over two.

And the y coordinate of the
midpoint is going to be the

average of the y coordinates.

So three plus negative one over two.

Let's see, negative five of plus seven is

positive two, over two is one.

Three minus one, three plus negative one,

that's positive two over two is one.

So the point one, comma, one
is a midpoint between these two

so one, comma, one, just like that.

So the line of reflection
is going to contain

these two points.

And two points define a line.

Let me draw the line of reflection,

just 'cause we did all of this work,

the line of reflection is
going to look something like,

I want to draw this a little
bit straighter than that,

it's going to look something like this.

And this makes sense that
this is a line of reflection.

I missed that magenta point a little bit,

so let me go through the magenta point.

Okay, there you go.

This makes sense that this
is a line of reflection

'cause you see that you
pick an arbitrary point

on segment ME,

say that point,

and if you reflected over this line.

This is the shortest
distance from the line.

You just go onto the other
side of the line equal distant

and you get to its corresponding point

on the image.

So it makes a lot of sense
that these are mirror images

if this is kind of the mirror here.

And you can image that this
is kind of the surface of

the water, if you're
looking at it at an angle.

I don't know if that helps you or not.

But anyway we found two points.

We found two points that
define that line of reflection

so now let's use the tool to type them in.

One is negative one, negative five.

The other is one, comma, one

so let me see if I can remember that.

I have a bad memory.

So one is negative one,
comma, negative five.

And then the other one is one, comma, one.

And we see it worked.

We see it worked.

When I did that, it actually
made the reflection happen

and notice it completely
went from this point

and now our blue is over the
image that we wanted to get to.

So we are done.