Point A is the original point before rotating. Which point represents A′ after A(4, 1) is rotated 90° counterclockwise: B, C, D, E, F, G, or H?
G
To find the point that represents A' after A(4, 1) is rotated 90° counterclockwise, we can use the rotation formula.
The formula for rotating a point (x, y) counterclockwise around the origin by θ degrees is:
x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)
In this case, we want to rotate A(4, 1) counterclockwise by 90°.
Using the rotation formula, we calculate:
x' = 4 * cos(90°) - 1 * sin(90°)
y' = 4 * sin(90°) + 1 * cos(90°)
cos(90°) = 0
sin(90°) = 1
x' = 4 * 0 - 1 * 1 = 0 - 1 = -1
y' = 4 * 1 + 1 * 0 = 4 + 0 = 4
Therefore, A' has the coordinates (-1, 4).
Among the given options B, C, D, E, F, G, and H, the point that represents A' is not provided.
To rotate a point counterclockwise by 90°, we need to swap the x- and y-coordinates and negate the new x-coordinate.
For point A(4, 1), swapping the x- and y-coordinates gives us A′(1, 4). Negating the new x-coordinate gives us A′(-1, 4).
Out of the given points B, C, D, E, F, G, and H, the only point that represents A′(-1, 4) after rotating A(4, 1) counterclockwise by 90° is D.