1. Break Down the Problem

We need to rotate the point $K(2, 5)$ by -90 degrees about the origin. This involves understanding the effect of rotation on Cartesian coordinates.

2. Relevant Concepts

The formula for rotating a point $(x, y)$ by an angle $\theta$ using matrix transformation is: [ \begin{pmatrix} x' \ y' \end{pmatrix}

\begin{pmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} ]

For a -90 degrees rotation:

• $\theta = -90^\circ$
• Thus, $\cos(-90^\circ) = 0$ and $\sin(-90^\circ) = -1$.

3. Analysis and Detail

Substituting the values into the rotation matrix: [ \begin{pmatrix} x' \ y' \end{pmatrix}

\begin{pmatrix} 0 & 1 \ -1 & 0 \end{pmatrix} \begin{pmatrix} 2 \ 5 \end{pmatrix} ]

Calculating the new coordinates: $x' = 0 \cdot 2 + 1 \cdot 5 = 5$ $y' = -1 \cdot 2 + 0 \cdot 5 = -2$

4. Verify and Summarize

The coordinates after a -90 degrees rotation result in the image of point $K$ being $K'(5, -2)$.

Final Answer

The coordinates of the image of point $K(2, 5)$ after a -90 degrees rotation about the origin are $K'(5, -2)$.