The image of the point (1, -2) after a rotation of 180° about the origin is
(-1,+2) by inspection
using 90 deg matrix
R90 =
|+0 -1|
|+1 +0|
R180 is R90 * R90
|+0 -1||+0 -1|
|+1 +0||+1 +0|
which is
|-1 +0|
|+0 -1|
|-1 +0| |+1|
|+0 -1| |-2|
= (-1, +2 )
To find the image of a point after a rotation of 180° about the origin, we can use the following steps:
1. Label the given point as (x, y). In this case, the point is (1, -2).
2. Apply the rotation formula:
- The x-coordinate of the image is given by x' = x * cos(180°) - y * sin(180°).
- The y-coordinate of the image is given by y' = x * sin(180°) + y * cos(180°).
3. Simplify the rotation formula:
- Since cos(180°) = -1 and sin(180°) = 0, the x-coordinate becomes x' = x * (-1) - y * 0.
- Similarly, the y-coordinate becomes y' = x * 0 + y * (-1).
4. Perform the calculations:
- For the x-coordinate: x' = 1 * (-1) - (-2) * 0 = -1.
- For the y-coordinate: y' = 1 * 0 + (-2) * (-1) = 2.
5. Write down the coordinates of the image:
The image of the point (1, -2) after a rotation of 180° about the origin is (-1, 2).
To find the image of a point after a rotation of 180° about the origin, we can use the following steps:
1. Given the point (1, -2), we need to find its new coordinates after a 180° rotation.
2. A 180° rotation about the origin means that the point will be reflected across the x-axis and the y-axis simultaneously.
3. Reflecting a point across the x-axis negates the y-coordinate, while reflecting a point across the y-axis negates the x-coordinate.
4. To reflect the point (1, -2) across the x-axis, we negate the y-coordinate, which gives us (1, 2).
5. Now, to reflect the point (1, 2) across the y-axis, we negate the x-coordinate, resulting in (-1, 2).
Therefore, the image of the point (1, -2) after a rotation of 180° about the origin is (-1, 2).