Question Video: Translating a Shape on a Grid | Nagwa Question Video: Translating a Shape on a Grid | Nagwa

Question Video: Translating a Shape on a Grid Mathematics • First Year of Preparatory School

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In the figure below, determine the coordinates of the points 𝐴′𝐡′𝐢′𝐷′, where 𝐴′𝐡′𝐢′𝐷′ is a translation of 𝐴𝐡𝐢𝐷 by (βˆ’2, βˆ’5).

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Video Transcript

In the figure below, determine the coordinates of the points 𝐴 prime, 𝐡 prime, 𝐢 prime, and 𝐷 prime, where 𝐴 prime 𝐡 prime 𝐢 prime 𝐷 prime is a translation of 𝐴𝐡𝐢𝐷 by negative two, negative five.

A translation is a transformation which moves a shape up or down and from side to side. A translation of negative two, negative five means the shape moves two places to the left and five places down. Seen as we’re translating a shape with four points, we have four coordinates to translate. There are a couple of ways we can find the images of each point after the translation. But let’s start by doing this graphically.

We’ll start with point 𝐴. If we move the point 𝐴 two units to the left and five units down, we find that the image of 𝐴, 𝐴 prime, is at the point six, one. Let’s do the same with point 𝐡. We move point 𝐡 two units to the left and five units down. And we find that the image of 𝐡, 𝐡 prime, is at the point six, negative four. Now let’s translate the point 𝐢. We move the point 𝐢 two units to the left and five units down. And we find that the image of 𝐢, 𝐢 prime, is at the point negative one, negative four. Finally, let’s translate the point 𝐷. We move the point 𝐷 two units to the left and five units down. And we find that the image of 𝐷, 𝐷 prime, is at the point one, negative one. Then, joining all our points, we see the translated shape 𝐴 prime 𝐡 prime 𝐢 prime 𝐷 prime.

Now we can read the coordinates of the translated points directly from the graph to get our answer. But let’s also see a different method we could have used. For a translation with horizontal displacement of π‘Ž units and a vertical displacement of 𝑏 units, the point π‘₯, 𝑦 maps to π‘₯ add π‘Ž, 𝑦 add 𝑏. Here are the coordinates of the four points we were given in the question: 𝐴, 𝐡, 𝐢, and 𝐷. We can use the formula above to map each point to its image. The horizontal displacement is negative two, so π‘Ž is negative two. The vertical displacement is negative five, so 𝑏 is negative five.

So we calculate the image of each of these coordinates by adding negative two to the π‘₯-coordinate and negative five to the 𝑦-coordinate, which then allows us to calculate the images of each of these points. That is, 𝐴 prime is six, one; 𝐡 prime is six, negative four; 𝐢 prime is negative one, negative four; and 𝐷 prime is one, negative one. Then, we can check these points by looking at the translation we carried out on the graph. And we see that the points match up.

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