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 (𝑥, 𝑦) → (−𝑥, −𝑦). 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 negative 𝑥, negative 𝑦. 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 negative 𝑥, negative 𝑦 to each set of coordinates to find the coordinates of the image. We note that the vertices of the image will be 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. Substituting this into the rule, we get 𝐴 prime with coordinates negative two, negative four. 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 negative three, zero. And finally, for point 𝐶, we use 𝑥 equal to zero and 𝑦 equal to zero. Substituting this into the rule returns the same coordinates: zero, zero. So 𝐶 prime remains at the origin.

Now that we have found the coordinates of the vertices of the image, we can plot them on an 𝑥𝑦-coordinate plane. We note that because the 𝑥-coordinate and 𝑦-coordinate are negative, 𝐴 prime must be in the third quadrant. Both options (a) and (b) show 𝐴 prime in the third quadrant, with coordinates negative two, negative four, and 𝐶 prime at the origin. But it is option (a) that shows 𝐵 prime at negative three, zero. By joining up the vertices with edges, we obtain the following sketch of triangle 𝐴 prime 𝐵 prime 𝐶 prime. Therefore, we conclude that the sketch of the image of triangle 𝐴𝐵𝐶 matches option (a).

We note that the transformation of triangle 𝐴𝐵𝐶 is a rotation, specifically a rotation of 180 degrees about the origin. In fact, we can remember that this transformation rule will always give a 180-degree rotation about the origin.