### 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.