Question

ΔABC with vertices A(-3, 0), B(-2, 3), C(-1, 1) is rotated 180° clockwise about the origin. It is then reflected across the line y = -x. What are the coordinates of the vertices of the image?

Answer 1

The soundest answer to this at an elementary can achieved by using graphing paper, plus tracing paper for the rotation. You also need to know that reflecting in y = x simply involves switching x for y and vice versa. It's explained here:

http://www.bbc.co.uk/schools/gcsebitesize/maths/geometry/transformationsrev4.shtml

For a more elevated approach, and I have no idea where you are on your journey, we can manipulate co-ordinates using matrices.

The rotation matrix is:

\( R(\theta )={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}\)

As the rotation is 180, it matters not BTW whether it is CW or CCW :)

Here R(180) is:

\( R={\begin{bmatrix} -1 & 0 \\ 0 &-1 \\\end{bmatrix}}\)

Reflection across y = x is simply switching co-ordinates so the matrix is:

\( M={\begin{bmatrix} 0 & 1 \\ 1 & 0 \\\end{bmatrix}}\)

If we combine them in the correct order we get:

\(RM = {\begin{bmatrix} -1 & 0 \\ 0 &-1 \\\end{bmatrix}} {\begin{bmatrix} 0 & 1 \\ 1 & 0 \\\end{bmatrix}} = {\begin{bmatrix} 0 & -1 \\ -1 & 0 \\\end{bmatrix}}\)

We can then apply that to all of your color-coded vertices as:

\( {\begin{bmatrix} 0 & -1 \\ -1 & 0 \\\end{bmatrix}} {\begin{bmatrix} \color{blue}{3} & \color{red}{-2} & -1 \\ \color{blue}{0} & \color{red}{3} & 1 \\\end{bmatrix}} = {\begin{bmatrix} \color{blue}{0} & \color{red}{-3} & -1 \\ \color{blue}{-3} & \color{red}{2} & 1 \\\end{bmatrix}}\)

Answer 2

though i still think tracing paper and a compass are the best answers....but hard to help with online