Question Video: Translating a Line Segment Given the Direction and Magnitude of the Translation

Scalars , vectors and directed line segment |senior 1 | geometry |مستر محمد خضر
Scalars , vectors and directed line segment |senior 1 | geometry |مستر محمد خضر

Video Transcript

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).

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