Question

What is the sequences of transformation that maps ABC to A'B'C'. A(1,2)B(2,5)C(5,1) to A'(3,-2)B'(4,-5)C'(7,-1)

Answer 1

• reflection across the x-axis
• translation to the right 2 units

Explanation:

Please refer to the attached figure.

The red triangle represents a reflection of ΔABC across the x-axis. Then the green triangle, ΔA'B'C' is a translation of the red one to the right by 2 units.

Of course, the sequence of reflection and translation can be reversed (doing the translation to the right first).

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Comment on the transformation

You can tell that at least one reflection is involved because the clockwise ordering of vertices A, B, C is reversed in the image. Such a reversal occurs when there have been an odd number of reflections.

The perpendicular bisectors of AA', BB', and CC' are all different, so the figure has not just been reflected once over a single line. This fact led us to consider that both reflection and translation were involved.

Algebraically, if you do the matrix operations to find the transformation involved, you will discover it to be ...

(x, y) ⇒ (x+2, -y) . . . . . translation right 2 units and reflection across the x-axis