Transformations in geometry are like dance moves for shapes. You can slide, flip, spin, or resize them. When you combine these moves, the order matters. It's like following a recipe - the steps you take change the final result.
Composing transformations lets you create complex changes with simple steps. You can make shapes travel, reflect, rotate, and scale in interesting ways. Understanding how these moves work together helps you predict and control the outcome of your geometric choreography.
- Involves applying a sequence of transformations to a geometric figure
- Order of transformations affects the final result
- Transformations are performed from right to left (last transformation listed is applied first)
- Common transformations that can be composed:
- Translations shift the figure without changing its orientation (sliding)
- Reflections flip the figure across a line (mirror image)
- Rotations turn the figure around a point (spinning)
- Dilations enlarge or shrink the figure (scaling)
- Translations
- Composing translations results in a single translation
- Resulting translation vector is found by adding the individual translation vectors (tip-to-tail method)
- Reflections
- Composing two reflections across parallel lines yields a translation
- Translation distance is twice the distance between the parallel lines
- Composing two reflections across intersecting lines yields a rotation
- Rotation angle is twice the angle between the intersecting lines
- Rotation center is the intersection point of the lines
- Rotations
- Composing rotations with the same center results in a single rotation
- Resulting rotation angle is the sum of the individual rotation angles
- Positive angles for counterclockwise rotations, negative for clockwise
- Dilations
- Composing dilations with the same center results in a single dilation
- Resulting scale factor is the product of the individual scale factors
- Scale factors greater than 1 enlarge the figure, between 0 and 1 shrink the figure
- Apply transformations from right to left
- Translations: Add translation vector components to each point's coordinates
- $(x, y) \rightarrow (x + a, y + b)$, where $(a, b)$ is the translation vector
- Reflections: Use reflection formulas based on the line of reflection
- Across x-axis: $(x, y) \rightarrow (x, -y)$
- Across y-axis: $(x, y) \rightarrow (-x, y)$
- Across line $y = x$: $(x, y) \rightarrow (y, x)$
- Across line $y = -x$: $(x, y) \rightarrow (-y, -x)$
- Rotations: Use rotation formulas based on the angle of rotation (counterclockwise around origin)
- 90°: $(x, y) \rightarrow (-y, x)$
- 180°: $(x, y) \rightarrow (-x, -y)$
- 270°: $(x, y) \rightarrow (y, -x)$
- Dilations: Multiply each point's coordinates by the scale factor
- $(x, y) \rightarrow (kx, ky)$, where $k$ is the scale factor
- Observe changes in shape, size, and orientation after each transformation
- Determine if the resulting figure is congruent (same size and shape) or similar (same shape, different size) to the original
- Translations, reflections, and rotations preserve congruence
- Dilations create similar figures, unless scale factor is 1
- Look for symmetries or patterns that emerge from multiple transformations
- Composing two reflections across perpendicular lines results in a half-turn (180° rotation)
- Composing two dilations with reciprocal scale factors $k$ and $\frac{1}{k}$ results in the original figure
- Compare coordinates of the original and transformed figures to find relationships
- Translations shift coordinates by constant amounts
- Reflections change the signs of coordinates
- Rotations cycle coordinates and may change signs
- Dilations multiply coordinates by the scale factor