# Question Video: Sketching the Image of a Triangle after a Geometric Transformation Mathematics

The given figure shows a triangle on the coordinate plane. Sketch the image of the triangle after the geometric transformation that maps (𝑥, 𝑦) → (𝑥 + 3, 𝑦 − 1). Which of the following matches your sketch? [A] Option a [B] Option b [C] Option c [D] Option d

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### Video Transcript

The given figure shows a triangle on the coordinate plane. Sketch the image of the triangle after the geometric transformation that maps 𝑥, 𝑦 onto 𝑥 plus three, 𝑦 minus one. Which of the following matches your sketch?

Looking at the given figure, we can see that the vertices of the triangle are 𝐴 with coordinates two, four; 𝐵 with coordinates three, zero; and 𝐶 with coordinates zero, zero. Having found the vertices, we can then apply the rule for the geometric transformation that maps 𝑥, 𝑦 onto 𝑥 plus three, 𝑦 minus one to each set of coordinates to find the coordinates of the image. The image will have vertices 𝐴 prime, 𝐵 prime, and 𝐶 prime.

For point 𝐴, we use 𝑥 equal to two and 𝑦 equal to four, since they are the 𝑥-coordinate and the 𝑦-coordinate for 𝐴. By Substituting this into the rule, we get 𝐴 prime with coordinates five, three. For point 𝐵, we use 𝑥 equal to three and 𝑦 equal to zero. Substituting this into the rule then gives us the coordinates of 𝐵 prime, which are six, negative one. And finally, for point 𝐶, we use 𝑥 equal to zero and 𝑦 equal to zero. Substituting this into the rule gives us the coordinates of 𝐶 prime: three, negative one.

Now that we have found the coordinates of the vertices of the image, we can plot them on the coordinate plane. Doing so gives us the following. Lastly, we join up the vertices with edges to obtain a sketch of the image, triangle 𝐴 prime 𝐵 prime 𝐶 prime. Comparing this with our options, we can see that this is the same as the sketch in option (c). We note that the transformation of triangle 𝐴𝐵𝐶 is a translation, specifically horizontally three units to the right and vertically one unit down.