Identifying a Translation

Triangle $ABC$ has vertices $A(1, 2)$, $B(4, 2)$, and $C(2, 5)$. After a transformation, the new triangle $A'B'C'$ has vertices $A'(4, 5)$, $B'(7, 5)$, and $C'(5, 8)$. What type of transformation is this?

Compare the coordinates: $A(1, 2) \to A'(4, 5)$: moved right 3 and up 3 $B(4, 2) \to B'(7, 5)$: moved right 3 and up 3 $C(2, 5) \to C'(5, 8)$: moved right 3 and up 3 = Each point moved the same amount

Check for consistent movement: All points shifted by $(+3, +3)$ = Same direction and distance

Identify the transformation: When every point moves the same distance in the same direction, it's a slide = Translation

Answer: This is a **translation** of 3 units right and 3 units up, written as $(x, y) \to (x + 3, y + 3)$

Recognizing a Reflection

Point $P(3, 4)$ is transformed to $P'(-3, 4)$. What type of transformation occurred and over what line?

Compare the coordinates: $P(3, 4) \to P'(-3, 4)$ $x$-coordinate: $3 \to -3$ (changed sign) $y$-coordinate: $4 \to 4$ (stayed the same) = Only $x$ changed

Analyze the change: The $x$-coordinate became its opposite while $y$ stayed the same = $(x, y) \to (-x, y)$

Identify the line of reflection: When $x$ changes sign and $y$ stays the same, the reflection is over the $y$-axis = Reflection over $y$-axis

Answer: This is a **reflection** over the $y$-axis. The rule is $(x, y) \to (-x, y)$

Understanding Rotation

Point $A(2, 0)$ is rotated $90°$ counterclockwise about the origin. What are the coordinates of $A'$?

Recall the rotation rule: For $90°$ counterclockwise rotation about the origin: $(x, y) \to (-y, x)$ = Apply the rule

Substitute the coordinates: $A(2, 0)$: $x = 2$, $y = 0$ $(-y, x) = (-0, 2) = (0, 2)$ = $A' = (0, 2)$

Verify the result: Distance from origin: $\sqrt{2^2 + 0^2} = 2$ Distance of $A'$: $\sqrt{0^2 + 2^2} = 2$ Distances are equal, as expected in rotation = Confirmed

Answer: After a $90°$ counterclockwise rotation about the origin, $A(2, 0)$ maps to $A'(0, 2)$

Applying Dilation

Triangle $PQR$ has vertices $P(2, 4)$, $Q(6, 4)$, and $R(4, 8)$. Find the coordinates of the image after a dilation with center at the origin and scale factor $k = \frac{1}{2}$.

Apply the dilation rule: Dilation with center at origin: $(x, y) \to (kx, ky)$ Scale factor $k = \frac{1}{2}$ = Multiply each coordinate by $\frac{1}{2}$

Transform each vertex: $P(2, 4) \to P'(2 \times \frac{1}{2}, 4 \times \frac{1}{2}) = P'(1, 2)$ $Q(6, 4) \to Q'(6 \times \frac{1}{2}, 4 \times \frac{1}{2}) = Q'(3, 2)$ $R(4, 8) \to R'(4 \times \frac{1}{2}, 8 \times \frac{1}{2}) = R'(2, 4)$ = All coordinates halved

Interpret the result: Since $k = \frac{1}{2} < 1$, the image is smaller than the pre-image The triangle shrinks toward the origin = Reduction

Answer: The image vertices are $P'(1, 2)$, $Q'(3, 2)$, and $R'(2, 4)$. The triangle is half the size of the original.

Mistake: Confusing reflection over the $x$-axis with reflection over the $y$-axis

Why: Students mix up which coordinate changes sign. Over the $x$-axis, $y$ changes sign. Over the $y$-axis, $x$ changes sign.

Correct: Remember: Reflection over $x$-axis: $(x, y) \to (x, -y)$. Reflection over $y$-axis: $(x, y) \to (-x, y)$. The axis that doesn't move keeps its coordinate the same.

Mistake: Rotating clockwise instead of counterclockwise (or vice versa)

Why: Positive angles typically mean counterclockwise rotation, but students often rotate the wrong direction.

Correct: Think of a clock: counterclockwise goes against the clock hands. For $90°$ counterclockwise: $(x, y) \to (-y, x)$. For $90°$ clockwise: $(x, y) \to (y, -x)$.

Mistake: Adding the scale factor instead of multiplying in dilations

Why: Dilation requires multiplication, not addition. A scale factor of 2 means multiply by 2, not add 2.

Correct: For dilation with scale factor $k$: $(x, y) \to (kx, ky)$. If $k = 2$, then $(3, 4) \to (6, 8)$, not $(5, 6)$.

Mistake: Thinking all transformations change the size of the figure

Why: Only dilation changes size. Translations, reflections, and rotations preserve size and shape (they are rigid transformations).

Correct: Translations, reflections, and rotations create congruent images. Dilations create similar images (same shape, different size).

Video Game Character Movement

Every time a video game character walks, jumps, or moves, the game uses translations to update the character's position on the screen.

If a character at position $(5, 2)$ moves 3 units right and 2 units up, their new position is $(5 + 3, 2 + 2) = (8, 4)$.

Mirror Reflections

When you look in a mirror, you see a reflection of yourself. Your left hand appears on the right side of your reflection, just like reflecting over a line.

If you stand 2 meters from a mirror, your reflection appears to be 2 meters on the other side - the same distance from the mirror line.

Ferris Wheel Rotations

As a Ferris wheel turns, each seat rotates around the center of the wheel. This is a real-world example of rotation about a fixed point.

If your seat starts at the bottom and the wheel rotates $90°$, you'll be on the side of the wheel.

Photo Editing and Zooming

When you zoom in on a photo or resize an image, you're applying a dilation. The image gets larger or smaller while maintaining its proportions.

Zooming a photo to 200% is like applying a dilation with scale factor $k = 2$. A 100-pixel image becomes 200 pixels.

Q: What's the difference between congruent and similar figures?

A: Congruent figures have the same shape AND size - they're identical copies. Similar figures have the same shape but may be different sizes. Translations, reflections, and rotations create congruent figures. Dilations create similar figures.

Q: Does the order of transformations matter?

A: Yes! Applying transformations in different orders often gives different results. For example, translating then reflecting may give a different image than reflecting then translating.

Q: What is a rigid transformation?

A: A rigid transformation (also called isometry) preserves distance and angle measures. Translations, reflections, and rotations are rigid transformations. Dilations are NOT rigid because they change size.