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. Calculate the angle measure in radians. Since the rotation angle is given as 45º, we need to convert it to radians by multiplying it by π/180. Hence, the angle in radians is 45 * π/180 = π/4.
2. To rotate a point (x, y) counterclockwise by an angle θ about the origin (0, 0), you can use the following rotation formulas:
x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)
3. Apply the rotation formulas to each coordinate of point R(2, 5) using the angle of π/4:
x' = 2 * cos(π/4) - 5 * sin(π/4)
y' = 2 * sin(π/4) + 5 * cos(π/4)
4. Simplify the expressions:
x' = 2 * (√2/2) - 5 * (√2/2)
= (√2) - (5√2)/2
= (-3√2)/2
y' = 2 * (√2/2) + 5 * (√2/2)
= (√2) + (5√2)/2
= (7√2)/2
5. Hence, the coordinates of R' after rotating triangle PQR counterclockwise about the origin 45º are approximately:
R'((-3√2)/2, (7√2)/2)
Note: The coordinates are given in terms of the square root of 2 (√2) to maintain the exactness of the answer.