# Problem of the Week Problem C and Solution Transformational Moves

## Problem

The three points $$A(1,1)$$, $$B(1,4)$$, and $$C(2,1)$$ are the vertices of $$\triangle ABC$$.
We perform transformations to the triangle, as follows. First, we shift $$\triangle ABC$$ to the right $$4$$ units. Then, we reflect the image in the $$x$$-axis. Then, we reflect the new image in the $$y$$-axis. Finally, we shift the newest image up $$5$$ units.

What are the coordinates of the vertices of the final triangle?

## Solution

In the solution we are going to use notation that is commonly used in transformations. When we transform point $$A$$, we label the transformed point as $$A'$$. We call this “$$A$$ prime”. When we transform point $$A'$$, we label the transformed point as $$A''$$. We call this “$$A$$ double prime”. This can continue for all four transformations and for vertices $$B$$ and $$C$$ as well.

When $$\triangle ABC$$ is shifted to the right $$4$$ units, the $$x$$-coordinate of each vertex increases by $$4$$. Thus, $$\triangle A'B'C'$$ has vertices $$A'(5,1)$$, $$B'(5,4)$$, and $$C'(6,1)$$.

When $$\triangle A'B'C'$$ is reflected in the $$x$$-axis, we multiply the $$y$$-coordinate of each vertex by $$-1$$. Thus, $$\triangle A''B''C''$$ has vertices $$A''(5,-1)$$, $$B''(5,-4)$$, and $$C''(6,-1)$$.

When $$\triangle A''B''C''$$ is reflected in the $$y$$-axis, we multiply the $$x$$-coordinate of each vertex by $$-1$$. Thus, $$\triangle A'''B'''C'''$$ has vertices $$A'''(-5,-1)$$, $$B'''(-5,-4)$$, and $$C'''(-6,-1)$$.

When $$\triangle A'''B'''C'''$$ is shifted up $$5$$ units, the $$y$$-coordinate of each vertex increases by $$5$$. Thus, $$\triangle A''''B''''C''''$$ has vertices $$A''''(-5,4)$$, $$B''''(-5,1)$$, and $$C''''(-6,4)$$.

Thus, the final triangle has vertices $$A''''(-5,4)$$, $$B''''(-5,1)$$ and $$C''''(-6,4)$$.