Video Transcript
In this video, we will learn how to
translate points, line segments, and shapes given the direction and magnitude of the
translation.
A translation can be thought of as
sliding an object a fixed distance in a fixed direction. For example, we can translate a
square of side length one centimeter to the right by two centimeters, as shown. We call the new position of the
square its image after the translation. Both squares are the same size and
orientation. The translation only affects the
position of the object.
We say that 𝐴 prime is the image
of point 𝐴 under the translation 𝑥 units in the direction of the ray 𝐵𝐶 if the
length 𝐴𝐴 prime is equal to 𝑥, the rays 𝐴𝐴 prime and 𝐵𝐶 are parallel and have
the same direction. We can translate line segments and
polygons by translating their endpoints or vertices.
Translations do not affect
lengths. So, if the line segment 𝐴𝐵 is
translated to the line segment 𝐴 prime 𝐵 prime, then the length of 𝐴𝐵 is equal
to the length of 𝐴 prime 𝐵 prime. Translations also do not affect
directions. In particular, if the line segment
𝐴𝐵 is translated to the line segment 𝐴 prime 𝐵 prime, then the line segment 𝐴𝐵
is parallel to the line segment 𝐴 prime 𝐵 prime. Similarly, the ray 𝐴𝐵 will be
translated to the ray 𝐴 prime 𝐵 prime.
Let’s take a look at an example of
determining the correct translation of a point in a given direction.
The image of point 𝑋 is 𝑋
prime following a translation of magnitude 𝑀𝑁 in the direction of the ray
𝑀𝑁. Which of the following diagrams
represents this?
Starting with the point in
space 𝑋, we are told that the translation is of magnitude 𝑀𝑁. This means that the image of
𝑋, 𝑋 prime, will lie somewhere on the circle of radius 𝑀𝑁 centered at the
original point 𝑋. We also have the ray 𝑀𝑁,
which points in the direction of the translation. We can therefore construct a
ray starting from the point 𝑋 that is parallel and in the same direction to the
ray 𝑀𝑁. And where this new ray touches
the circle will be where the image 𝑋 prime lies.
To do this, we start by
constructing a ray that passes through 𝑋 in the direction of 𝑀. We seek to duplicate the angle
𝑁𝑀𝑋 at the point 𝑋. We can do this by constructing
two arcs of circles of equal radius centered at the points 𝑋 and 𝑀, labeling
the points of intersection between the circle centered at 𝑀 and the two rays 𝐴
and 𝐵. We then measure this distance
between 𝐴 and 𝐵 and draw another arc of another circle centered on the far
point of intersection between the circle arc centered at 𝑋 and the ray passing
through 𝑋 and 𝑀. A ray passing through the point
𝑋 and the point of intersection of these two arcs will be parallel to the ray
𝑀𝑁.
We can therefore draw the ray
starting at the point 𝑋 and passing through this point of intersection. And the point of intersection
between this ray and the circle will be the image of the point 𝑋 prime. Looking at the possible
answers, we can see that this matches with the diagram in 𝐴.
Let’s now look at an example where
we will determine which of five given diagrams shows the correct translation of a
line segment.
The image of the line segment
𝐶𝐷 is the line segment 𝐶 prime 𝐷 prime following a translation of magnitude
𝐴𝐵 in the direction of the ray 𝐴𝐵. Which of the following diagrams
represents this?
We can approach this problem
with a process of elimination. Recall that translating a line
segment is equivalent to translating its endpoints, in this case 𝐶 and 𝐷, to
their images 𝐶 prime and 𝐷 prime and connecting the images of the points with
a new line segment. In all five diagrams, the ray
𝐴𝐵 is pointing up and to the left. So any translation of the line
segment that doesn’t place the image in this direction must be incorrect. This immediately rules out
answers (A) and (B).
Next, the translation is of
magnitude 𝐴𝐵, meaning the distance between one endpoint of the line segment
and its image must be the length 𝐴𝐵. In answer (D), the distance
between the points 𝐶 and its image 𝐶 prime is clearly less than the length
𝐴𝐵. So the answer cannot be
(D).
And finally translations leave
the lengths of line segments unchanged. And in answer (E), we can see
that the length of the image line segment 𝐶 prime 𝐷 prime is greater than the
length of the original line segment 𝐶𝐷. Therefore, the answer cannot be
(E).
Therefore, the only remaining
answer is (C). And we can see that the line
segment is indeed translated in the correct direction by the correct distance
and does not affect the size and orientation of the line.
We can check if this is correct
by constructing the translation. We’ve cleared the incorrect
answers to give some space. Taking the line segment 𝐶𝐷,
we construct two rays in the same direction as the ray 𝐴𝐵, one starting at 𝐶
and the other starting at 𝐷. Next, we set the radius of a
compass equal to the length 𝐴𝐵 and trace circles centered at 𝐶 and 𝐷. We label the points of
intersection between the circles and the rays 𝐶 prime and 𝐷 prime. 𝐶 prime and 𝐷 prime are a
distance 𝐴𝐵 from 𝐶 and 𝐷, respectively, and in the direction of the ray
𝐴𝐵. So they are the images of 𝐶
and 𝐷 under the translation. Therefore, the line segment
joining the points 𝐶 prime and 𝐷 prime is the image of the line segment 𝐶𝐷
under the translation. And this matches visually with
option (C).
Let’s now look at an example of how
to determine the correct translation of a line segment in a square grid.
Fill in the blank. In the figure, 𝐴𝐵𝐶𝐷 is a
square, where all interior squares are congruent and 𝐴𝑀 is equal to one
centimeter. Then, the image of the line
segment 𝐻𝑌 by a translation of magnitude two centimeters in the direction of
the ray 𝐿𝐵 is what.
To start with, the line segment
𝐻𝑌 is this one here and the ray 𝐿𝐵 is this ray here. Translating a line segment is
equivalent to translating its endpoints, in this case 𝐻 and 𝑌, and then
connecting the images of these points with a new line segment. This means we can answer this
question by finding the images of the endpoints of this line segment under the
translation.
All of the squares are
congruent and have a side length of one centimeter. The sides of the squares in the
left-to-right direction are also parallel. Therefore, the translation of
magnitude two centimeters in the direction of the ray 𝐿𝐵 is equivalent to the
ray 𝐻𝐸. Therefore, the image of 𝐻
under the translation 𝐻 prime is the point 𝐸. Similarly, the image of the
point 𝑌 under the translation 𝑌 prime is the point 𝑁. Therefore, the image of the
line segment 𝐻𝑌 by a translation of magnitude two centimeters in the direction
of the ray 𝐿𝐵 is the line segment 𝐸𝑁.
So far, we have applied
translations to given geometric objects. However, it is also possible to
determine the magnitude and direction of a translation by using a point and its
image under the translation.
For example, imagine we are given a
point 𝐴 and its image under a translation, 𝐴 prime. The ray 𝐴𝐴 prime is in the
direction of the translation, since this is the direction between 𝐴 and its image
𝐴 prime. Likewise, the magnitude of this
translation must be the length 𝐴𝐴 prime, since this is the distance 𝐴 is
translated. This leads to the following
property. If the point 𝐴 is translated on 𝐴
prime by a translation, then the translation has magnitude equal to the length 𝐴𝐴
prime and in the direction of the ray 𝐴𝐴 prime.
In the final example, we will use
this property to determine the magnitude and direction of the translation of a given
triangle onto its image.
Fill in the blank. In the figure, the triangles
𝑋𝐿𝑀, 𝐿𝑌𝑁, 𝑁𝑀𝐿, and 𝑀𝑁𝑍 are congruent. The triangle 𝐿𝑌𝑁 is the
image of the triangle 𝑀𝑁𝑍 by a translation of magnitude what in the direction
of what.
The magnitude of a translation
is equivalent to the distance between a point and its image under the
translation. And the direction of a
translation can be described by the ray from a point passing through its
image. The whole triangle 𝑀𝑁𝑍 can
be translated by translating its vertices. Therefore, we can determine the
magnitude and direction of the translation of the triangle by finding the image
under the translation of just one of its vertices.
So we can pick a vertex, for
example, 𝑀, and determine its image under the translation. The two triangles 𝑀𝑁𝑍 and
𝐿𝑌𝑁 are highlighted here as the orange and purple triangles,
respectively. Translations do not affect the
scale or orientation of a shape. Therefore, the relative
positions of the vertices in the triangle will remain the same. The top vertex will remain at
the top, and so on.
Therefore, the top vertex of
the image triangle 𝐿 is the image of the top vertex of the original triangle
𝑀. The magnitude of the
translation is therefore given by the distance between 𝑀 and 𝐿, 𝑀𝐿. And the direction is given by
the direction of the ray starting at the vertex 𝑀 and passing through 𝐿,
𝑀𝐿. Therefore, the answer is
(A).
Let’s finish this video by going
over some key points. A translation of an object can be
thought of as sliding the object in space without changing its shape, size, or
orientation. We say that 𝐴 prime is the image
of point 𝐴 under the translation 𝑋 units in the direction of the ray 𝐵𝐶 if 𝐴𝐴
prime is equal to 𝑋 and the rays 𝐴𝐴 prime and 𝐵𝐶 have the same direction.
Translations do not affect the
length of line segments or their direction. In particular, if the line segment
𝐴𝐵 is translated onto the line segment 𝐴 prime 𝐵 prime, then the length 𝐴𝐵 is
equal to the length 𝐴 prime 𝐵 prime and the line segments 𝐴𝐵 and 𝐴 prime 𝐵
prime are parallel. And finally, we can find the
magnitude and direction of a translation by finding a point and its image under the
translation and determining the distance between these points and the direction of
the ray from the point to its image.
