Video Transcript
Three points 𝐴, 𝐵, and 𝐶
with coordinates one, three; one, two; and four, one, respectively, are
reflected in the 𝑥-axis to the points 𝐴 prime, 𝐵 prime, and 𝐶 prime. Determine the coordinates of 𝐴
prime, 𝐵 prime, and 𝐶 prime. Is the measure of angle 𝐴𝐵𝐶
less than, greater than, or equal to the measure of angle 𝐴 prime 𝐵 prime 𝐶
prime?
We recall first that a
reflection in the 𝑥-axis maps a point with coordinates 𝑥, 𝑦 to the point with
coordinates 𝑥, negative 𝑦. The 𝑥-coordinate stays the
same, and the 𝑦-coordinate is multiplied by negative one. We can then apply this
transformation to each point separately. Point 𝐴, which has coordinates
one, three, is mapped to the point one, negative three. Point 𝐵 with coordinates one,
two is mapped to one, negative two. And point 𝐶 with coordinates
four, one is mapped to four, negative one.
To answer the second part of
the question, it may be helpful to sketch the points 𝐴, 𝐵, and 𝐶 together
with their images 𝐴 prime, 𝐵 prime, and 𝐶 prime on a coordinate grid. The angles we’re interested in
are angle 𝐴𝐵𝐶 and angle 𝐴 prime 𝐵 prime 𝐶 prime, which are marked on the
figure. We can see that these are both
obtuse angles, which appear to be of equal measure. If we recall that reflections
map a geometric figure to a congruent geometric figure, then we can deduce that
angles 𝐴𝐵𝐶 and 𝐴 prime 𝐵 prime 𝐶 prime must be of equal measure, as they
are corresponding angles in congruent triangles.
So we’ve completed the
problem. We’ve found the coordinates of
𝐴 prime, 𝐵 prime, and 𝐶 prime and determined that the two angles are of equal
measure.
