### Video Transcript

In this lesson, we will learn how
to reflect points and shapes in given reflection lines. There are many different ways we
can transform an object without altering its shape or size. Weβll focus on one of these
transformations now. This one is called a
reflection.

Reflections can be thought of as
mirror images. When we look in a mirror, we see a
virtual image of ourselves and our surroundings, as if we were on the other side of
the mirror. In geometry, we reflect objects in
a mirror line. And we call the result of the
reflection an image. A particularly important feature of
reflections is that the real object and its image must be the same distance from the
mirror line. Weβve experienced this when moving
towards or away from a mirror. The reflected image of ourselves
moves as well.

Suppose that we wanted to reflect a
single point π΄ through a mirror line to give the image, point π΄ prime. If we did so, we would notice that
the mirror line is the perpendicular bisector of the line segment between these
points. We can use this to define the
reflection of any object more formally. A reflection is a transformation
that preserves the perpendicular distances of all points from the mirror line.

Letβs now look at an example of how
to apply this definition to identify the image of a reflection.

Five points are plotted on the
grid in the figure below. When reflecting point π across
the dotted line, which of the four other points will it map onto? Is it option (A) point π,
option (B) point π, (C) point π, or (D) point π
?

Remember, a reflection is a
transformation that preserves the perpendicular distances of all points from the
mirror line. In particular, if we define the
image of π to be the point π prime, this is the reflection through the mirror
line where the mirror line is the perpendicular bisector of the line segment
that joins these points.

So, we can find the image of π
by drawing a line perpendicular to the mirror line. Thatβs the dotted line in this
picture. When we do, we see that this
perpendicular line passes through point π
. In particular, point π
is the
same perpendicular distance from the dotted line as point π. So, the correct answer is
option (D). Point π
is the image of π in
a reflection through the dotted line.

It is worth noting that we can
apply the definition of reflection to more than just points. For instance, a straight line is a
geometric object consisting of all points that satisfy a given rule. So, we can reflect these points
through a line, which will result in the reflection of the object. In particular, we might reflect a
shape by individually reflecting the vertices and then joining these together. Letβs demonstrate these ideas in
the next two examples.

Which of the following
represents the image of the line segment π΄π΅ after a reflection in the line
πΏ?

Remember, we can reflect a line
segment through a line by reflecting its endpoints. When we do so, the image of
each point will be the same perpendicular distance from the mirror line as the
original point. So, letβs do this for each
diagram. When we do so, we can see that
options (B) and (C) are definitely incorrect. And at first glance, option (D)
looks correct. But we can see that the
vertices have been reversed. In fact, the correct answer is
option (A). This represents the image of
the line segment π΄π΅ after a reflection in line πΏ.

Letβs now look at a similar example
involving a polygon.

Which of the following
represents the image of triangle π΄π΅πΆ after a reflection in the line π?

Remember, when we reflect a
single point, we preserve the perpendicular distance of this point from the
mirror line. This means we can reflect a
polygon by reflecting each of its vertices and then joining these together. Take, for instance, option
(A). If we reflect vertex π΅ in the
mirror line, we see that its image does not correspond to the image in the
second triangle. We have a similar issue with
the diagram in option (B).

However, when we reflect vertex
π΅ in option (C), it corresponds to point π΅ prime in the second triangle. Reflecting point π΄ through the
mirror line corresponds to the point π΄ prime in the second triangle. And when we complete the same
process for point πΆ, we do indeed see that this second triangle is a full
reflection through the mirror line. Itβs worth noting that for
option (B) the triangle did look like it might be correct, but the vertices were
in the wrong order. So, we have confirmed that the
correct answer is option (C).

Before we consider another example,
letβs write down some useful properties of reflections. First, as we observed in our
earlier example, reflecting a line segment will give a congruent line segment. In other words, it will give
another line segment of the exact same size. We also know that the mirror line
is the perpendicular bisector of any line segment between a point and its image. By considering both of these two
points, we can now see that reflecting the shape gives a congruent shape. And in particular, the measure of
any angles are preserved.

Letβs now look at how to apply
these properties to determine the length of the sides and the angle measures in a
triangle.

In the following figure,
triangle π΄ prime π΅ prime πΆ prime is the image of triangle π΄π΅πΆ by
reflection in the line πΏ. (1) Fill in the blanks. The length of π΄ prime πΆ prime
equals blank centimeters, and the length of π΄ prime π΅ prime equals blank
centimeters. (2) Fill in the blanks. Line segment π΄π΄ prime is
blank to line segment π΅π΅ prime, and line segment πΆπΆ prime is blank to line
πΏ. (3) Find the measure of angle
π΄.

Remember, when we reflect a
polygon in a mirror line, we create a second congruent polygon. This means that the two
triangles in our diagram are congruent. That in turn means that their
line segments and angle measures are equal. This fact helps us to answer
part (1). Line segment π΄πΆ is congruent
to line segment π΄ prime πΆ prime. They must have the same
lengths. Since line segment π΄πΆ is four
centimeters, line segment π΄ prime πΆ prime must also be four centimeters. And we put four in the first
blank space.

Next, line segment π΄π΅ must be
congruent to line segment π΄ prime π΅ prime. And so, π΄ prime π΅ prime must
be six centimeters in length. And six goes in our second
blank space.

Letβs now consider question
(2). First, we add line segments
π΄π΄ prime and π΅π΅ prime to the diagram. We know that these line
segments must be perpendicular to the mirror line. If theyβre both perpendicular
to the mirror line, we can conclude some further information. That is, their alternate angles
are equal, and they must in fact be parallel to one another. To find the second blank word
in question (2), we add the line segment to the diagram. And of course, we know that
πΆπΆ prime is perpendicular to line πΏ.

Finally, we consider question
(3). Remember, these two triangles
are congruent, which means they share angle measures. In particular, this means that
the measure of angle π΄ must be equal to the measure of angle π΄ prime. Angle π΄ prime is 31 degrees,
so angle π΄ is also 31 degrees.

And so, we have filled in the
blanks. The correct entries were four,
six, parallel, perpendicular, and 31 degrees.

It is worth noting that we can
reflect any shape through a line by reflecting all of the points. And this includes circles, where we
would reflect its center and maintain its radius.

Given a circle with center π
that intersects with line πΏ at points π΄ and π΅, draw an image of circle π
after a reflection in line πΏ. Which of the following
statements is correct? Is it option (A) line segment
π΄π is parallel to line segment π΄π prime? Option (B) line segment π΅π is
parallel to line segment π΅π prime. Option (C) line segment ππ
prime is perpendicular to line segment π΄π΅. Option (D) ππ prime is equal
to π΄π΅. Or option (E) π΄ prime π΅ prime
is greater than π΄π΅.

To reflect a circle in a mirror
line, we must first reflect its center and then preserve its radius. We reflect the center, thatβs
point π, by first creating the perpendicular line to πΏ that passes through
π. Next, we know that line segment
ππ΄ is a radius of our original circle. So, we can trace an arc with
center π΄ and with the same radius. The point where this arc
intersects our line is the center of our image. So, we have the image of our
circle after reflection. We can now use this to identify
the correct statement.

Thereβs no way that π΄π and
π΄π prime can be parallel to one another. They quite clearly meet and
form an acute angle. In fact, π΅π and π΅π prime
cannot be parallel either for the same reasons. Of course, we do know that ππ
prime must be perpendicular to π΄π΅. And this is because we
constructed the perpendicular line bisector of line πΏ at the very start. And line πΏ passes through
π΄π΅. The correct answer is option
(C). Line segment ππ prime is
perpendicular to line segment π΄π΅.

Weβll now recap the key points from
this lesson. First, a reflection is a
transformation which preserves the perpendicular distances of all points from the
mirror line. Next, we learned that when we
reflect a point through a mirror line and then join these two points, the mirror
line will be the perpendicular line bisector of that newly created line segment.

We saw that we can reflect a line
segment by individually reflecting its endpoints. And to reflect a polygon, we
reflect its vertices and join them together. We learned that when we reflect a
shape, its image preserves its length and angles. So, its image is in fact congruent
to the original shape. And finally, we learned that we can
reflect a circle by reflecting its center and preserving its radius.