### Video Transcript

In this video, we will learn how to
translate points, line segments, and shapes on the coordinate plane.

Translations are one of the
fundamental ways of moving geometrical objects without changing their shape. In fact, translating an object can
be thought of as sliding the object in space without changing its size or its
orientation. Translations are defined by two
properties: firstly their magnitude, which tells us how far the object needs to
move, and secondly their direction, which tells us which way the object is
moving. When thinking about the direction
of a translation, we describe how far the object has to move horizontally and how
far it has to move vertically. For example, we may say that a
translation moves an object four units to the right and two units up.

On this coordinate plane, we have a
point π΄, which has the coordinates three, four. If we move this point one unit to
the right and then two units up, it ends up at the point four, six. We call this point the image of π΄
after the given translation and denote it using the notation π΄ prime. We could also find the image of
point π΄ following this translation without drawing on a coordinate grid if we
consider the effect the translation has on the coordinates of a point.

Moving a point one unit to the
right will increase its π₯-coordinate by one. And moving a point two units up
will increase its π¦-coordinate by two. So we could say that under this
translation of moving a point one unit right and two units up, the point with
coordinates π₯, π¦ is mapped to the point with coordinates π₯ plus one, π¦ plus
two. So point π΄, which had coordinates
three, four, is mapped to the point π΄ prime, which has coordinates three plus one,
four plus two, which is indeed the point with coordinates four, six.

If we want to translate objects to
the left or down, then we describe the displacement using negative values. For example, a translation three
units to the left would correspond to subtracting three from the π₯-coordinate. So the horizontal displacement is
negative three. A translation of six units down
would correspond to subtracting six from the π¦-coordinate. So the vertical displacement is
negative six.

We can define translations using
mapping notation more formally as follows. If a translation in the coordinate
plane has a horizontal displacement of π units and a vertical displacement of π
units, then the point with coordinates π₯, π¦ will be mapped to the point with
coordinates π₯ plus π, π¦ plus π. And we can write this using the
mapping notation of an arrow. The signs of π and π tell us the
direction of the displacement: positive for to the right or up and negative for to
the left or down. Letβs now consider an example of
applying this definition to find the coordinates of the image of a point under a
given translation.

Find the coordinates of the image
of the point 13, four under the translation π₯, π¦ is mapped to π₯ plus five, π¦
minus two.

Weβve been given this
transformation algebraically. We can recall that the given
mapping notation means that this translation has a horizontal displacement of five
units and a vertical displacement of negative two units. In other words, the translation
maps each point five units to the right and two units down. Itβs two units down because we are
subtracting two from the π¦-coordinate.

Applying this mapping to the point
with coordinates 13, four gives the point with coordinates 13 plus five, four minus
two, which is the point with coordinates 18, two. We could also apply this
transformation graphically if we had access to squared paper on which to draw a
coordinate grid. Hereβs the point with coordinates
13, four. To add five to the π₯-coordinate,
we move five units to the right. And then to subtract two from the
π¦-coordinate, we move two units down. The image of the point 13, four
following this translation is indeed the point with coordinates 18, two.

Letβs now consider another example
in which weβll rewrite a translation in terms of the horizontal and vertical
displacements it represents.

Which of the following is
equivalent to a translation of π₯, π¦ is mapped to π₯ plus two, π¦ minus three? (A) A translation of two units
right and three units up. (B) A translation of two units left
and three units down. (C) A translation of two units
right and three units down. (D) A translation of three units
right and two units up. Or (E) a translation of three units
right and two units down.

Weβve been given this translation
in the form of a mapping, and we want to determine what it means in words. We can recall that, in general, the
translation represented by the mapping π₯, π¦ is mapped to π₯ plus π, π¦ plus π
has a horizontal displacement of π units and a vertical displacement of π
units. The signs of π and π tell us the
direction in which the translation occurs.

Looking at the mapping here, the
value of π is positive two and the value of π is negative three. So the horizontal displacement is
positive two, which means two units to the right. The vertical displacement is
negative three. And as we are subtracting three
from the π¦-coordinate, this means three units down. Looking carefully at the five
options we were given, we can see that a translation of two units right and three
units down is option (C).

Another way of describing
translations is as a movement along a ray. For example, letβs consider how we
can translate a point πΆ π΄π΅ units in the direction π΄π΅. Here, the direction of the
translation is expressed as the direction of the line connecting π΄ to π΅. And the magnitude is the length of
the line segment π΄π΅ itself. We do this by first drawing in a
ray starting at point πΆ that is parallel to the line segment π΄π΅ and in the same
direction. We then mark on this ray the point
that is π΄π΅ units away from πΆ so that π΄π΅ is the same length as πΆπΆ prime.

Without squared paper, we can
construct this parallel ray using a compass and a ruler. And we can find the position of the
image of point πΆ on this ray by using compasses with the length set to the length
of the line segment π΄π΅. This is one method which allows us
to find the image of point πΆ following translation. However, we can also think about
this translation in terms of the horizontal and vertical displacements.

To travel along the ray from π΄ to
π΅, we need to move three units to the left and one unit down. So, to translate the point πΆ π΄π΅
units in the direction of the ray π΄π΅, we can also move the point πΆ three units to
the left and then one unit down. Letβs now consider an example in
which we translate a triangle in the coordinate plane by using the coordinates of
its three vertices.

List the coordinates π΄ prime, π΅
prime, and πΆ prime that represent the image of triangle π΄π΅πΆ after translation
with magnitude ππ in the direction of the ray ππ, where π has coordinates one,
three and π has coordinates four, five, given that π΄ has coordinates five, three;
π΅ has coordinates one, two; and πΆ has coordinates three, six.

Letβs start by rewriting the
translation in terms of how it affects the π- and π-coordinates. And to do this, weβll begin by
sketching π and π. A translation of magnitude ππ in
the direction of the ray ππ is equivalent to the translation that maps the point
π to the point π, since π is ππ units away from π in the direction of the ray
ππ. From our diagram, we can see that
this translation will have the effect of increasing the π-coordinate by three and
increasing the π-coordinate by two. So we can express this translation
as a point with coordinates lowercase π₯, lowercase π¦ is mapped to the point π₯
plus three, π¦ plus two.

We can then apply this mapping to
each vertex of the triangle. For the point π΄, which has
coordinates five, three, it will be mapped to the point five plus three, three plus
two. So π΄ prime has coordinates eight,
five. The point π΅ with coordinates one,
two will be mapped to the point with coordinates one plus three, two plus two. So π΅ prime has coordinates four,
four. Finally, for vertex πΆ, the point
three, six, this is mapped to the point three plus three, six plus two. So πΆ prime has coordinates six,
eight.

We can check our answer by plotting
the points π΄, π΅, and πΆ together with the image points π΄ prime, π΅ prime, and πΆ
prime on the coordinate plane. We can see that each point has
indeed been translated three units to the right and two units up. So we have that the coordinates of
π΄ prime are eight, five; the coordinates of π΅ prime are four, four; and the
coordinates of πΆ prime are six, eight.

Looking at the two triangles from
the previous example, we can see that they are congruent. That is, they are exactly the same
shape and size. This highlights some useful
properties of translations. Firstly, the size of an object is
preserved under translations. In particular, since the size is
preserved, the lengths of line segments remain constant under translations. So, for example, we could say that
the line segment π΅πΆ is the same length as the line segment π΅ prime, πΆ prime. Secondly, the orientation of a
shape is preserved under translation. This means that a line and its
image under translation will be parallel. So, for example, the line segment
π΅π΄ is parallel to the line segment π΅ prime, π΄ prime. Thirdly, translations preserve the
measure of any angle. So the measure of the angle π΅π΄πΆ
is the same as the measure of the angle π΅ prime π΄ prime πΆ prime. And the same is true for the other
two angles in our triangles.

Letβs now consider another example
in which weβll find the image of a point on the coordinate plane under a translation
whose magnitude and direction are given in terms of a ray between two given
points.

The following translation π΄π΅ is
equivalent to a horizontal displacement from one to five and a vertical displacement
from four to two. Find the image of point πΆ by
performing translation π΄π΅ in the direction of the ray π΄π΅.

Weβre told that this translation
π΄π΅, which is the mapping needed to map point π΄ onto point π΅, is equivalent to a
horizontal displacement from one to five and a vertical displacement from four to
two. So we can calculate both the
horizontal and vertical displacements. The horizontal position increases
by four units, and the vertical position decreases by two units. We can express this translation in
mapping notation as the point π₯, π¦ is mapped to the point π₯ plus four, π¦ minus
two.

From the figure, we can determine
that the point πΆ has coordinates one, two. So, applying this mapping, point πΆ
will be mapped to the point with coordinates one plus four, two minus two, which is
the point five, zero. We can check that this answer is
correct by applying the translation graphically. Since the translation maps π΄ to
π΅, point πΆ must be mapped the same distance and direction as point π΄. So we can draw a ray starting at
point πΆ, which is the same length as the line segment π΄π΅ and in the direction of
the ray π΄π΅, to find the point πΆ prime. This confirms that the coordinates
of πΆ prime, which is the image of point πΆ following translation π΄π΅ in the
direction of the ray π΄π΅, are five, zero.

Weβll now consider one final
example in which weβll apply a given transformation to three points on the
coordinate plane.

Three points β π΄ one, negative
five; π΅ two, negative five; and πΆ two, four β are translated by the mapping π₯, π¦
is mapped to π₯ minus three, π¦ plus one to points π΄ prime, π΅ prime, and πΆ
prime. Determine π΄ prime, π΅ prime, and
πΆ prime.

We recall first that the notation
π₯, π¦ is mapped to π₯ plus π, π¦ plus π describes the translation that maps point
π₯, π¦ to the point with coordinates π₯ plus π and π¦ plus π. It corresponds to a translation
with a horizontal displacement of π units and a vertical displacement of π
units. For the translation in this
question, we have π equal to negative three and π equal to positive one. We are decreasing the π₯-coordinate
by three and increasing the π¦-coordinate by one.

To find the image of each point
under this translation, we can substitute the π₯- and π¦-coordinates of each point
into the map. So, for the point π΄, which has
coordinates one, negative five, its image will be the point with coordinates one
minus three, negative five plus one. Thatβs the point negative two,
negative four. π΅, which has coordinates two,
negative five, will be mapped to the point with coordinates two minus three,
negative five plus one. So π΅ prime has coordinates
negative one, negative four. And finally, point πΆ, which has
coordinates two, four, will be mapped to the point with coordinates two minus three,
four plus one, which is the point negative one, five.

So we determined π΄ prime, π΅
prime, and πΆ prime. We could also consider this
translation graphically. Here are the points π΄, π΅, and πΆ
represented on a coordinate plane. A horizontal displacement of
negative three units means weβre decreasing the π₯-coordinate of each point by
three. And a vertical displacement of
positive one unit means weβre increasing the π¦-coordinate of each point by one. This confirms that the coordinates
of π΄ prime, π΅ prime, and πΆ prime are negative two, negative four; negative one,
negative four; and negative one, five, respectively.

Letβs now summarize the key points
from this video. Firstly, any translation in the
coordinate plane can be thought of in terms of the horizontal and vertical
displacements. In general, a translation in the
coordinate plane with a horizontal displacement of π units and a vertical
displacement of π units can be expressed as the mapping π₯, π¦ is mapped to π₯ plus
π, π¦ plus π. If π is positive, the horizontal
displacement is to the right, whereas if π is negative, the horizontal displacement
is to the left. If π is positive, the vertical
displacement is up, and if π is negative, the vertical displacement is down. We also saw that an object and its
image under translation are congruent, which also reveals the following
properties. Lengths of line segments are
preserved under translations. A line and its image under
translation are parallel. And finally, angle measure is
preserved under translations.