Q37. - Algebra 2

Question

GYMNASTICS

The drawing at the right shows four positions of a man performing the giant swing in the high bar event. Suppose this drawing is placed on a coordinate grid with hand grips at $H\left(0,0\right)$ and the toe of the figure in the upper right corner at $T\left(7,8\right)$. Find the coordinates of the toes of the other three figures, if each successive figure has been rotated $90°$ counter clockwise about the origin.

Step-by-Step Solution

Answer

Coordinates of upper left corner is $\left(-8,7\right)$, lower left corner is $\left(-7,-8\right)$ and lower right corner is $\left(8,-7\right)$

Step 1 - Vertex matrix

An ordered pair $\left(x,y\right)$ is represented by a column matrix $\left[\begin{array}{c}x\\ y\\ \end{array}\right]$ which is known as vertex matrix of an ordered pair. So vertex matrix of ordered pair $\left(7,8\right)$ is $\left[\begin{array}{c}7\\ 8\\ \end{array}\right]$.

Step 2 - Coordinate after first rotation

Vertex matrix after rotation of $90°$ counter clockwise first time gives the coordinate of upper left corner. This matrix is product of matrix $\left[\begin{array}{cc}0& -1\\ 1& 0\\ & \end{array}\right]$ with vertex matrix $\left[\begin{array}{c}7\\ 8\\ \end{array}\right]$ as shown below

$\left[\begin{array}{cc}0& -1\\ 1& 0\\ & \end{array}\right]\times \left[\begin{array}{c}7\\ 8\\ \end{array}\right]=\left[\begin{array}{c}-8\\ 7\\ \end{array}\right]$

As, each coordinate is represented by a column of vertex matrix, so each column represents one coordinate.

Thus, coordinates of upper left corner point is $\left(-8,7\right)$

Step 3 - Coordinate after second rotation

Vertex matrix after second rotation of $90°$ counter clockwise gives the coordinate of lower left corner. This matrix is product of matrix $\left[\begin{array}{cc}0& -1\\ 1& 0\\ & \end{array}\right]$ with vertex matrix $\left[\begin{array}{c}-8\\ 7\\ \end{array}\right]$ as shown below

$\left[\begin{array}{cc}0& -1\\ 1& 0\\ & \end{array}\right]\times \left[\begin{array}{c}-8\\ 7\\ \end{array}\right]=\left[\begin{array}{c}-7\\ -8\\ \end{array}\right]$

As, each coordinate is represented by a column of vertex matrix, so each column represents one coordinate.

Thus, coordinates of lower left corner point is $\left(-7,-8\right)$

Step 4 - Coordinate after third rotation

Vertex matrix after rotation of $90°$ counter clockwise third time gives the coordinate of lower right corner. This matrix is product of matrix $\left[\begin{array}{cc}0& -1\\ 1& 0\\ & \end{array}\right]$ with vertex matrix $\left[\begin{array}{c}-7\\ -8\\ \end{array}\right]$ as shown below

$\left[\begin{array}{cc}0& -1\\ 1& 0\\ & \end{array}\right]\times \left[\begin{array}{c}-7\\ -8\\ \end{array}\right]=\left[\begin{array}{c}8\\ -7\\ \end{array}\right]$

As, each coordinate is represented by a column of vertex matrix, so each column represents one coordinate.

Thus, coordinates of lower right corner point is $\left(8,-7\right)$