What is the image of the point (4,5) after a rotation of 90 degree counterclockwise about the origin?
such a rotation takes (x,y)→(-y,x)
so plug in your numbers and review the topic
Well, after a 90-degree counterclockwise rotation, the point (4,5) will become the star of the show! It will dazzle the crowd and transform into the point (-5,4). Talk about a stunning performance, folks!
To find the image of a point after a rotation of 90 degrees counterclockwise about the origin, you can use the following rotation matrix:
\[
\begin{bmatrix}
\cos(\theta) & -\sin(\theta) \\
\sin(\theta) & \cos(\theta)
\end{bmatrix}
\]
Here, \(\theta\) represents the angle of rotation (90 degrees in this case).
Let's substitute the values into the rotation matrix:
\[
\begin{bmatrix}
\cos(90) & -\sin(90) \\
\sin(90) & \cos(90)
\end{bmatrix}
\]
Evaluating the trigonometric functions:
\[
\begin{bmatrix}
0 & -1 \\
1 & 0
\end{bmatrix}
\]
Now, we can multiply the rotation matrix with the coordinate of the point (4,5):
\[
\begin{bmatrix}
0 & -1 \\
1 & 0
\end{bmatrix}
\begin{bmatrix}
4 \\
5
\end{bmatrix}
\]
Performing the matrix multiplication:
\[
\begin{bmatrix}
0 \cdot 4 + (-1) \cdot 5 \\
1 \cdot 4 + 0 \cdot 5
\end{bmatrix}
\]
Simplifying the multiplication:
\[
\begin{bmatrix}
-5 \\
4
\end{bmatrix}
\]
Therefore, the image of the point (4,5) after a rotation of 90 degrees counterclockwise about the origin is (-5,4).
To find the image of a point after a rotation of 90 degrees counterclockwise about the origin, you can use the following steps:
1. Plot the given point (4, 5) on a Cartesian coordinate plane. The x-coordinate represents the horizontal distance from the origin, and the y-coordinate represents the vertical distance.
2. To rotate a point counterclockwise by 90 degrees about the origin, we need to swap the x and y coordinates and negate the new x-coordinate.
- The new x-coordinate, let's call it x', is equal to the original y-coordinate negated: x' = -y = -5.
- The new y-coordinate, let's call it y', is equal to the original x-coordinate: y' = x = 4.
3. Plot the new point (-5, 4) on the same Cartesian coordinate plane.
The image of the point (4, 5) after a rotation of 90 degrees counterclockwise about the origin is (-5, 4).