Final answer:
The image of point A(1,4) after a 90-degree counter-clockwise rotation about the origin is A'(-4,1). Additionally, the distance to the origin remains constant during rotations.
Step-by-step explanation:
The question asks to determine the image of a point after a 90-degree counter-clockwise rotation around the origin. The transformation of point A(1,4) will result in a new point A'(x',y'). To perform a 90-degree counter-clockwise rotation, we can apply the following rules: x' = -y and y' = x. Therefore, for point A(1,4), after the rotation, the new coordinates will be A'(-4,1).
Another essential concept covered by this question is the idea that the distance of a point to the origin is invariant under rotations. This means that no matter how you rotate a point around the origin, the distance to the origin does not change. Mathematically, this is represented as the sum of the squares of the coordinates remaining the same (x² + y² = constant).
To determine the image of point A(1,4) after a 90-degree counterclockwise rotation about the origin A'(1,-4), we can use the rotation matrix. The rotation matrix for a 90-degree counterclockwise rotation is:
| cos(90°) -sin(90°) | | x | | -y |
| sin(90°) cos(90°) | * | y | = | x |
Plugging in the coordinates of point A into the rotation matrix gives us:
| 0 -1 | | 1 | | -4 |
| 1 0 | * | 4 | = | 1 |
The image of point A(1,4) after the rotation is A'(-4,1).