Given point A(2, 6) on the triangle, where would the new point on the triangle be after a rotation of 90 degrees clockwise?%0D%0A%0D%0A(1 point

Answer 1

To find the new point after a rotation of 90 degrees clockwise, we can use the rotation matrix formula:

\[\begin{bmatrix}x'\\ y'\end{bmatrix} = \begin{bmatrix}\cos(\theta) & \sin(\theta)\\ -\sin(\theta) & \cos(\theta)\end{bmatrix} \begin{bmatrix}x\\ y\end{bmatrix}\]

Where \(\theta\) is the angle of rotation, which is 90 degrees in this case. The point A(2, 6) can be represented as the column vector \(\begin{bmatrix}2\\ 6\end{bmatrix}\). Substituting these values into the rotation matrix formula, we get:

\[\begin{bmatrix}x'\\ y'\end{bmatrix} = \begin{bmatrix}\cos(90^\circ) & \sin(90^\circ)\\ -\sin(90^\circ) & \cos(90^\circ)\end{bmatrix} \begin{bmatrix}2\\ 6\end{bmatrix}\]

Calculating the values, we have:

\[\begin{bmatrix}x'\\ y'\end{bmatrix} = \begin{bmatrix}0 & 1\\ -1 & 0\end{bmatrix} \begin{bmatrix}2\\ 6\end{bmatrix}\]

Multiplying the matrices, we get:

\[\begin{bmatrix}x'\\ y'\end{bmatrix} = \begin{bmatrix}0(2) + 1(6)\\ -1(2) + 0(6)\end{bmatrix} = \begin{bmatrix}6\\ -2\end{bmatrix}\]

Therefore, the new point after a 90-degree clockwise rotation is B(6, -2).