What is the equation of the line Y=3X-2 rotated 60 degrees clockwise about the point (2,5)?
PLEASE HELP
To rotate a point or a line in a coordinate system, you can use a mathematical technique called rotation transformation. In this case, we need to rotate the line Y=3X-2 about the point (2,5) by 60 degrees clockwise.
To accomplish this, we can follow these steps:
Step 1: Translate the line and the center of rotation to the origin.
- Subtract 2 from X and 5 from Y to shift the line and center of rotation to the origin.
- The equation becomes Y' = 3(X'-2)-2, where (X', Y') represents the translated coordinates.
Step 2: Perform the rotation.
- Apply the rotation transformation matrix to rotate the translated line.
- The rotation transformation matrix for a clockwise rotation by angle theta is:
| cos(theta) -sin(theta) |
| sin(theta) cos(theta) |
In this case, theta is 60 degrees.
Step 3: Translate the line and center of rotation back to their original positions.
- Add 2 to X' and 5 to Y' to shift the line and center of rotation back to their original positions.
- The equation becomes Y = 3(X'-2)+5, where (X', Y') represents the transformed coordinates.
Now, let's simplify the equation step by step:
1. Translate the line and center of rotation to the origin:
Y' = 3(X'-2) - 2
2. Apply the rotation transformation matrix with theta = 60 degrees:
X'' = X' * cos(60) - Y' * sin(60)
Y'' = X' * sin(60) + Y' * cos(60)
3. Translate the line and center of rotation back to their original positions:
X = X'' + 2
Y = Y'' + 5
By substituting the transformed values into the original equation of the line, we get:
Y = 3(X-2) + 5
Thus, the equation of the line Y=3X-2 rotated 60 degrees clockwise about the point (2,5) is Y = 3(X-2) + 5.