Given point A (2, 6 on the triangle, where would the new point on the triangle be after a rotation of 90 degrees clockwise
To find the new coordinates after a rotation of 90 degrees clockwise, we can use the rotation matrix:
\[\begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}\]
Here, θ (theta) is the angle of rotation.
In this case, we have a rotation of 90 degrees clockwise, so θ = -90 degrees (or -π/2 radians).
The coordinates of point A are (2,6), so we can substitute these values into the formula:
\[\begin{bmatrix} \cos(-90^\circ) & -\sin(-90^\circ) \\ \sin(-90^\circ) & \cos(-90^\circ) \end{bmatrix} \begin{bmatrix} 2 \\ 6 \end{bmatrix}\]
Using the trigonometric values of -90 degrees, we have:
\[\begin{bmatrix} \cos(-90^\circ) & -\sin(-90^\circ) \\ \sin(-90^\circ) & \cos(-90^\circ) \end{bmatrix} \begin{bmatrix} 2 \\ 6 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \begin{bmatrix} 2 \\ 6 \end{bmatrix}\]
Carrying out the matrix multiplication:
\[\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \begin{bmatrix} 2 \\ 6 \end{bmatrix} = \begin{bmatrix} (0 \cdot 2) + (1 \cdot 6) \\ (-1 \cdot 2) + (0 \cdot 6) \end{bmatrix} = \begin{bmatrix} 6 \\ -2 \end{bmatrix}\]
Therefore, the new coordinates after a rotation of 90 degrees clockwise would be (6,-2).
To find the new coordinates after a clockwise rotation of 90 degrees, we can use the following formula:
x' = x * cosθ - y * sinθ
y' = x * sinθ + y * cosθ
In this case, the original coordinates of point A are (2, 6) and we want to rotate it 90 degrees clockwise.
Step 1: Convert the angles to radians.
Since the rotation angle is given in degrees, we need to convert it to radians. 90 degrees is equal to θ = π/2 radians.
Step 2: Apply the rotation formula.
Using the formula, substitute the coordinates of point A and the value of θ into the equations.
x' = 2 * cos(π/2) - 6 * sin(π/2)
y' = 2 * sin(π/2) + 6 * cos(π/2)
Step 3: Simplify the equations.
cos(π/2) = 0 and sin(π/2) = 1.
x' = 2 * 0 - 6 * 1 = -6
y' = 2 * 1 + 6 * 0 = 2
Therefore, after a clockwise rotation of 90 degrees, the new coordinates of point A would be (-6, 2) on the triangle.