Part 1: The Rules of Transformations
1. Reflections (The "Flip")
A reflection flips a figure across a line (called the line of reflection) to create a mirror image.
Rule for reflecting across the x-axis:
(x, y) → (x, -y) (The y-coordinate changes sign)
Rule for reflecting across the y-axis:
(x, y) → (-x, y) (The x-coordinate changes sign)
Rule for reflecting across the line y=x:
(x, y) → (y, x) (The x and y coordinates switch)
Rule for reflecting across the line y=-x:
(x, y) → (-y, -x) (Switch x and y, AND change both signs)
Example: Reflect point B(3, 7) across the y-axis.
We use the rule (x, y) → (-x, y).
The x-coordinate '3' becomes '-3'. The y-coordinate '7' stays the same.
The new point, B', is at (-3, 7).
2. Rotations (The "Turn")
A rotation turns a figure about a fixed point (usually the origin).
Rule for 90° counter-clockwise rotation:
(x, y) → (-y, x) (Switch x and y, and change the sign of the new x)
Rule for 180° rotation:
(x, y) → (-x, -y) (Change the sign of both x and y)
Rule for 270° counter-clockwise rotation (or 90° clockwise):
(x, y) → (y, -x) (Switch x and y, and change the sign of the new y)
Rule for 360° rotation:
(x, y) → (x, y) (The figure returns to its original position)
Example: Rotate point C(6, 2) 180° about the origin.
We use the rule (x, y) → (-x, -y).
The x-coordinate '6' becomes '-6'. The y-coordinate '2' becomes '-2'.
The new point, C', is at (-6, -2).
Part 2: Practice Problems
For each problem, a triangle is given by the coordinates of its vertices (corners). Apply the transformation rule to find the new coordinates for A', B', and C'.
1. Reflection
Triangle ABC has vertices A(-2, 4), B(3, 1), C(0, 5).
Reflect the triangle across the x-axis.
New Coordinates:
A' = ( , )
B' = ( , )
C' = ( , )
2. Reflection
Triangle ABC has vertices A(2, -1), B(4, -5), C(-1, -3).
Reflect the triangle across the y-axis.
New Coordinates:
A' = ( , )
B' = ( , )
C' = ( , )
3. Reflection
Triangle ABC has vertices A(2, 8), B(5, 1), C(-3, 6).
Reflect the triangle across the line y=x.
New Coordinates:
A' = ( , )
B' = ( , )
C' = ( , )
4. Rotation
Triangle ABC has vertices A(7, 2), B(8, 5), C(1, 4).
Rotate the triangle 180° about the origin.
New Coordinates:
A' = ( , )
B' = ( , )
C' = ( , )
5. Rotation
Triangle ABC has vertices A(5, 8), B(1, 9), C(2, 3).
Rotate the triangle 90° counter-clockwise about the origin.
New Coordinates:
A' = ( , )
B' = ( , )
C' = ( , )