Even though I have not graphed these triangles, knowing the relationships is enough to answer these questions with sufficient justification:
a) When reflecting points over the x-axis, the following rule applies to all transformed points:
(x,y)--->(x,-y)
What this means is that the x-coordinate remains unchanged, but the y-coordinate changes to its opposite. Therefore, the coordinates of the new triangle is:
A'(-4,1), B'(-1,3), and C'(-6,2)
b) Yes, \(\triangle\)ABC and \(\triangle\)A'B'C' are similar. In fact, it is an isometry (congruent transformation). This is because a reflection is merely a mirror image of the pre-image.
c) For this tranformation, the rule is as follows:
(x,y)-->(-x,-y)
Therefore, the resulting coordinates become:
X(4,1), Y(1,3), Z(6,2)
Also, this transformation is the equivalent of a rotation about the origin 180 degrees counterclockwise. We know this because if the scale factor, k,<0 then it always results in a rotation.
d) Triangle PQR is similar to Triangle ABC because multiplying the coordinates by any scale factor always results in similar figures.
