Video Transcript
List the coordinates 𝐴 prime, 𝐵
prime, and 𝐶 prime that represent the image of triangle 𝐴𝐵𝐶 after translation
with magnitude 𝑋𝑌 in the direction of the ray 𝑋𝑌, where 𝑋 has coordinates one,
three and 𝑌 has coordinates four, five, given that 𝐴 has coordinates five, three;
𝐵 has coordinates one, two; and 𝐶 has coordinates three, six.
Let’s start by rewriting the
translation in terms of how it affects the 𝑋- and 𝑌-coordinates. And to do this, we’ll begin by
sketching 𝑋 and 𝑌. A translation of magnitude 𝑋𝑌 in
the direction of the ray 𝑋𝑌 is equivalent to the translation that maps the point
𝑋 to the point 𝑌, since 𝑌 is 𝑋𝑌 units away from 𝑋 in the direction of the ray
𝑋𝑌. From our diagram, we can see that
this translation will have the effect of increasing the 𝑋-coordinate by three and
increasing the 𝑌-coordinate by two. So we can express this translation
as a point with coordinates lowercase 𝑥, lowercase 𝑦 is mapped to the point 𝑥
plus three, 𝑦 plus two.
We can then apply this mapping to
each vertex of the triangle. For the point 𝐴, which has
coordinates five, three, it will be mapped to the point five plus three, three plus
two. So 𝐴 prime has coordinates eight,
five. The point 𝐵 with coordinates one,
two will be mapped to the point with coordinates one plus three, two plus two. So 𝐵 prime has coordinates four,
four. Finally, for vertex 𝐶, the point
three, six, this is mapped to the point three plus three, six plus two. So 𝐶 prime has coordinates six,
eight.
We can check our answer by plotting
the points 𝐴, 𝐵, and 𝐶 together with the image points 𝐴 prime, 𝐵 prime, and 𝐶
prime on the coordinate plane. We can see that each point has
indeed been translated three units to the right and two units up. So we have that the coordinates of
𝐴 prime are eight, five; the coordinates of 𝐵 prime are four, four; and the
coordinates of 𝐶 prime are six, eight.
