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

Question Video: Translating a Line Segment Given the Direction and Magnitude of the Translation Mathematics • First Year of Preparatory School

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The image of the line segment 𝐢𝐷 is the line segment 𝐢′𝐷′ following a translation of magnitude 𝐴𝐡 in the direction of the ray 𝐴𝐡. Which of the following diagrams represents this? [A] Option (A) [B] Option (B) [C] Option (C) [D] Option (D) [E] Option (E)

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Video Transcript

The image of the line segment 𝐢𝐷 is the line segment 𝐢 prime 𝐷 prime following a translation of magnitude 𝐴𝐡 in the direction of the ray 𝐴𝐡. 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 𝐢 and 𝐷, to their images 𝐢 prime and 𝐷 prime and connecting the images of the points with a new line segment. In all five diagrams, the ray 𝐴𝐡 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 𝐴𝐡, meaning the distance between one endpoint of the line segment and its image must be the length 𝐴𝐡. In answer (D), the distance between the points 𝐢 and its image 𝐢 prime is clearly less than the length 𝐴𝐡. 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 𝐢 prime 𝐷 prime is greater than the length of the original line segment 𝐢𝐷. 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 𝐢𝐷, we construct two rays in the same direction as the ray 𝐴𝐡, one starting at 𝐢 and the other starting at 𝐷. Next, we set the radius of a compass equal to the length 𝐴𝐡 and trace circles centered at 𝐢 and 𝐷. We label the points of intersection between the circles and the rays 𝐢 prime and 𝐷 prime. 𝐢 prime and 𝐷 prime are a distance 𝐴𝐡 from 𝐢 and 𝐷, respectively, and in the direction of the ray 𝐴𝐡. So they are the images of 𝐢 and 𝐷 under the translation. Therefore, the line segment joining the points 𝐢 prime and 𝐷 prime is the image of the line segment 𝐢𝐷 under the translation. And this matches visually with option (C).

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