Video Transcript
Three points π΄, π΅, and πΆ
with coordinates one, three; one, two; and four, one, respectively, are
reflected in the π₯-axis to the points π΄ prime, π΅ prime, and πΆ prime. Determine the coordinates of π΄
prime, π΅ prime, and πΆ prime. Is the measure of angle π΄π΅πΆ
less than, greater than, or equal to the measure of angle π΄ prime π΅ prime πΆ
prime?
We recall first that a
reflection in the π₯-axis maps a point with coordinates π₯, π¦ to the point with
coordinates π₯, negative π¦. The π₯-coordinate stays the
same, and the π¦-coordinate is multiplied by negative one. We can then apply this
transformation to each point separately. Point π΄, which has coordinates
one, three, is mapped to the point one, negative three. Point π΅ with coordinates one,
two is mapped to one, negative two. And point πΆ 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 π΄, π΅, and πΆ together
with their images π΄ prime, π΅ prime, and πΆ prime on a coordinate grid. The angles weβre interested in
are angle π΄π΅πΆ and angle π΄ prime π΅ prime πΆ 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 π΄π΅πΆ and π΄ prime π΅ prime πΆ prime must be of equal measure, as they
are corresponding angles in congruent triangles.
So weβve completed the
problem. Weβve found the coordinates of
π΄ prime, π΅ prime, and πΆ prime and determined that the two angles are of equal
measure.