Given a point (2,4) on a triangle in Quadrant I, how would you describe the coordinates of the new point when the triangle is rotated 90° clockwise?
To describe the coordinates of the new point when the triangle is rotated 90° clockwise, we can use the formula for the rotation of a point about the origin in the Cartesian coordinate system.
Let's assume the given point (2, 4) is labeled as A. To rotate this point 90° clockwise, we need to apply the following transformation:
x' = x * cos(θ) + y * sin(θ)
y' = -x * sin(θ) + y * cos(θ)
Where θ represents the angle of rotation.
Plugging in the values for point A (x = 2, y = 4) and θ = -90° (since we are rotating clockwise), the equations become:
x' = 2 * cos(-90°) + 4 * sin(-90°)
y' = -2 * sin(-90°) + 4 * cos(-90°)
Now, we evaluate the trigonometric functions for -90°:
cos(-90°) = 0
sin(-90°) = -1
Therefore, the new coordinates (x', y') of the point after rotating 90° clockwise become:
x' = 2 * 0 + 4 * (-1) = 0 - 4 = -4
y' = -2 * (-1) + 4 * 0 = 2 + 0 = 2
Hence, the new point after rotating the triangle 90° clockwise is (-4, 2).