The coordinates of point A on triangle ABC are (2, -4). What would be the image of point A after triangle ABC is reflected over the y-axis andthen rotated about the origin 180 degree?
Triangle ABC is shown on the graph. What are the coordinates of the image of point B after the triangle is rotated 270° about the origin?
To find the image of a point after a reflection over the y-axis and a rotation about the origin, we can follow these steps:
1. Reflection over the y-axis:
- When a point is reflected over the y-axis, the x-coordinate becomes its opposite and the y-coordinate remains unchanged.
- The coordinates of point A after the reflection over the y-axis would be (-2, -4).
2. Rotation about the origin 180 degrees:
- In a 2D space, a rotation of 180 degrees about the origin can be achieved by negating both the x-coordinate and the y-coordinate of the point.
- The coordinates of point A after the rotation would be (-(-2), -(-4)), which simplifies to (2, 4).
Therefore, the image of point A after triangle ABC is reflected over the y-axis and then rotated 180 degrees about the origin is (2, 4).
To find the image of point A after reflecting triangle ABC over the y-axis and rotating it 180 degrees about the origin, we can follow these steps:
Step 1: Reflecting over the y-axis
The reflection over the y-axis results in flipping the x-coordinate while keeping the y-coordinate the same. In this case, the x-coordinate of point A, which is 2, will change to -2 (-2, -4).
Step 2: Rotating 180 degrees about the origin
To rotate a point 180 degrees about the origin, we need to use the rotation matrix:
[ cosθ -sinθ ]
[ sinθ cosθ ]
Since we are rotating 180 degrees, θ will be π (pi).
Applying the rotation matrix, the new coordinates of point A after rotation will be:
-2 × cos(π) - (-4) × sin(π) , -2 × sin(π) + (-4) × cos(π)
Simplifying the expression, we get:
(-2 × (-1) - (-4) × 0) , (-2 × 0 + (-4) × (-1))
Which becomes:
(2, 4)
Therefore, the image of point A after the reflection over the y-axis and rotating 180 degrees about the origin is (2, 4).