Triangle PQR has vertices P(0, 1), Q(0, -4), and R(2, 5). Find the coordinates of R' to the nearest hundredth after rotating triangle PQR counterclockwise about the origin 45º.
To find the coordinates of R' after rotating triangle PQR counterclockwise about the origin 45º, you can follow these steps:
1. Find the coordinates of R relative to the origin. Since the origin is (0, 0), the coordinates of R are (2, 5).
2. Apply the rotation formula for counterclockwise rotation about the origin by an angle of 45º:
x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)
where (x, y) are the original coordinates, (x', y') are the coordinates after rotation, and θ is the angle of rotation.
3. Substitute the values into the rotation formulas:
x' = 2 * cos(45º) - 5 * sin(45º)
y' = 2 * sin(45º) + 5 * cos(45º)
4. Calculate the values:
x' = 2 * √2/2 - 5 * √2/2
y' = 2 * √2/2 + 5 * √2/2
5. Simplify the expressions:
x' = √2 - 5√2/2
y' = √2 + 5√2/2
6. To the nearest hundredth, approximate the values:
x' ≈ -1.92
y' ≈ 6.07
Therefore, the coordinates of R', after rotating triangle PQR counterclockwise about the origin 45º, are approximately (-1.92, 6.07).