### Video Transcript

The image of the line segment
πΆπ· is the line segment πΆ prime π· prime following a translation of magnitude
π΄π΅ in the direction of the ray π΄π΅. 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 πΆ and π·, to
their images πΆ prime and π· prime and connecting the images of the points with
a new line segment. In all five diagrams, the ray
π΄π΅ is pointing up and to the left. So any translation of the line
segment that doesnβt place the image in this direction must be incorrect. This immediately rules out
answers (A) and (B).

Next, the translation is of
magnitude π΄π΅, meaning the distance between one endpoint of the line segment
and its image must be the length π΄π΅. In answer (D), the distance
between the points πΆ and its image πΆ prime is clearly less than the length
π΄π΅. 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 πΆ prime π· prime is greater than the
length of the original line segment πΆπ·. 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. Weβve cleared the incorrect
answers to give some space. Taking the line segment πΆπ·,
we construct two rays in the same direction as the ray π΄π΅, one starting at πΆ
and the other starting at π·. Next, we set the radius of a
compass equal to the length π΄π΅ and trace circles centered at πΆ and π·. We label the points of
intersection between the circles and the rays πΆ prime and π· prime. πΆ prime and π· prime are a
distance π΄π΅ from πΆ and π·, respectively, and in the direction of the ray
π΄π΅. So they are the images of πΆ
and π· under the translation. Therefore, the line segment
joining the points πΆ prime and π· prime is the image of the line segment πΆπ·
under the translation. And this matches visually with
option (C).