Given triangle ABC with vertices at points A( - 1, 3 ), B( 1, - 1 ), and C( 4, 2 ) and if the triangle is rotated 90° clockwise about the point ( 4, 1 ), find the new coordinates of  C'.

To find the new coordinates of C' after a 90° clockwise rotation about the point (4,1), we first need to translate the point to the origin, perform the rotation, and then translate it back.

1. Translate the point to the origin: The original coordinates of C are (4,2). To translate this point to the origin, we subtract the coordinates of the point of rotation (4,1) from the coordinates of C. This gives us (4-4, 2-1) = (0,1).

2. Perform the rotation: A 90° clockwise rotation is equivalent to a 90° counterclockwise rotation in the negative direction. The formula for a 90° counterclockwise rotation is (x',y') = (-y,x). Applying this to our translated point (0,1) gives us (x',y') = (-1,0).

3. Translate the point back: Now we add the coordinates of the point of rotation (4,1) to the rotated point (-1,0). This gives us (4-1, 1+0) = (3,1).

So, the new coordinates of C' after a 90° clockwise rotation about the point (4,1) are (3,1).

To find the new coordinates of C' after a 90° clockwise rotation about the point (4,1), we first need to translate the point C to the origin by subtracting the rotation point from C.

The coordinates of C are (4,2). So, after translating to the origin, we get:

C_translated = C - rotation_point C_translated = (4-4, 2-1) = (0,1)

Now, we apply the rotation matrix for a 90° clockwise rotation. The rotation matrix for a 90° clockwise rotation is:

[0 1] [-1 0]

We multiply this matrix with the translated coordinates of C:

C_rotated = rotation_matrix * C_translated C_rotated = [00 + 11, -10 + 01] = (1,0)

Finally, we translate back to the original position by adding the rotation point to C_rotated:

C' = C_rotated + rotation_point C' = (1+4, 0+1) = (5,1)

So, the new coordinates of C' after a 90° clockwise rotation about the point (4,1) are (5,1).

To find the new coordinates of C' after a 90° clockwise rotation about the point (4,1), we first need to translate the point to the origin, perform the rotation, and then translate it back.

1. Translate the point to the origin: The original coordinates of C are (4,2). To translate this point to the origin, we subtract the coordinates of the point of rotation (4,1) from the coordinates of C. This gives us (4-4, 2-1) = (0,1).

2. Perform the rotation: A 90° clockwise rotation of a point (x,y) around the origin is given by the transformation (-y,x). Applying this to our translated point (0,1) gives us (-1,0).

3. Translate the point back: We now add the coordinates of the point of rotation (4,1) to the coordinates of our rotated point (-1,0). This gives us (4-1, 1+0) = (3,1).

So, the new coordinates of C' after a 90° clockwise rotation about the point (4,1) are (3,1).

To find the new coordinates of C' after a 90° clockwise rotation about the point (4,1), we can use the formula for 2D rotation:

The formula for a 90° clockwise rotation about the origin (0,0) is (x', y') = (y, -x). However, since we are rotating about the point (4,1) and not the origin, we first need to shift the point C to the origin by subtracting the rotation point from C.

The original coordinates of C are (4,2). Subtracting the rotation point (4,1) gives us (0,1).

Now we can apply the rotation formula. Substituting x = 0 and y = 1 into the formula (x', y') = (y, -x) gives us (1, 0).

Finally, we shift the point back to its original position by adding the rotation point (4,1) to the rotated point (1,0). This gives us the new coordinates of C' as (5,1).