Translations and Reflections

🔷 Translations and Reflections

Part 1 of 5 — Sliding Figures: Translations

Topics in This Part

🔑 Key Concept: A translation slides every point of a figure the same distance in the same direction — no turning, no flipping. The figure that comes out is identical in size and shape to the one that went in.

What Is a Transformation?

A transformation takes a figure and moves it to a new position on the coordinate plane.

• The original figure is the pre-image.
• The moved figure is the image.
• We label image points with a prime symbol: point $A$ moves to $A'$ (read "A prime").

A rigid motion (also called an isometry) is a transformation that keeps the figure the same size and same shape — it just relocates it. Translations and reflections are both rigid motions.

💡 Because translations and reflections never change size or shape, the pre-image and image are always congruent ($\cong$). We'll come back to this big idea in Part 5.

Concept Check 🎯

The Translation Rule

To translate a point, add a fixed amount to each coordinate:

$(x,\ y) \;\longrightarrow\; (x + a,\ y + b)$

• $a$ controls horizontal motion: $+a$ moves right, $-a$ moves left.
• $b$ controls vertical motion: $+b$ moves up, $-b$ moves down.

Example: translate $A(2, 3)$ right $4$ and up $1$

Right $4$ means $a = +4$; up $1$ means $b = +1$:

$A(2,\ 3) \;\longrightarrow\; A'(2 + 4,\ 3 + 1) = A'(6,\ 4)$

Example: translate $B(-1, 5)$ left $3$ and down $2$

Left $3$ means $a = -3$; down $2$ means $b = -2$:

$B(-1,\ 5) \;\longrightarrow\; B'(-1 - 3,\ 5 - 2) = B'(-4,\ 3)$

⚠️ Watch the signs. "Left" and "down" are the negative directions. Mixing them up is the #1 translation error.

Translate the Point 🧮

Apply each translation and enter the new coordinates.

1) Translate $(4, 1)$ right $2$, up $3$. New $x = \,?$ and new $y = \,?$ 2) Translate $(-5, 6)$ left $4$, down $6$. New $x = \,?$ and new $y = \,?$

Translating a Whole Figure

To translate a figure, apply the same rule to every vertex, then connect the new points in the same order.

Example: translate triangle $ABC$ by $(x, y) \to (x + 3,\ y - 2)$

Every vertex moves right $3$ and down $2$, so the whole triangle slides as one solid piece.

🔑 Key Idea: Side lengths and angles never change under a translation, so $\triangle ABC \cong \triangle A'B'C'$. You only ever need to move the vertices — the sides follow.

Concept Check 🎯

🔷 Translations and Reflections

Part 2 of 5 — Flipping Figures: Reflections Over the Axes

🔑 The Idea: A reflection flips a figure over a line called the line of reflection, producing a mirror image. Each point and its image are the same distance from that line, on opposite sides.

Reflecting Over the $x$-axis and $y$-axis

The two most common mirror lines are the axes. Each has a simple coordinate rule.

Over the $x$-axis (the horizontal axis)

$(x,\ y) \;\longrightarrow\; (x,\ -y)$

The $x$-coordinate stays put; the sign of $y$ flips. The figure flips up–down.

Over the $y$-axis (the vertical axis)

$(x,\ y) \;\longrightarrow\; (-x,\ y)$

The $y$-coordinate stays put; the sign of $x$ flips. The figure flips left–right.

💡 Memory hook: Reflect over the $x$-axis → the $x$ stays the same; reflect over the $y$-axis → the $y$ stays the same. The other coordinate gets negated.

Pick the Image 🔽

Choose the image of each point under the given reflection.

Worked Example: reflect $A(3, 5)$ over the $x$-axis

Keep $x$, negate $y$:

$A(3,\ 5) \;\longrightarrow\; A'(3,\ -5)$

$A$ is $5$ units above the $x$-axis; $A'$ is $5$ units below it — same distance, opposite side. ✓

Worked Example: reflect $B(-2, 4)$ over the $y$-axis

Negate $x$, keep $y$:

$B(-2,\ 4) \;\longrightarrow\; B'(2,\ 4)$

$B$ is $2$ units left of the $y$-axis; $B'$ is $2$ units right of it. ✓

⚠️ A point on the mirror line does not move. For example, $(0, 7)$ reflected over the $y$-axis stays at $(0, 7)$ because $-0 = 0$.

Concept Check 🎯

Distance From the Mirror

A handy check: a point and its image are equally far from the line of reflection.

Example

$C(5, 2)$ reflected over the $x$-axis becomes $C'(5, -2)$.

• $C$ is $2$ units above the $x$-axis.
• $C'$ is $2$ units below the $x$-axis.

Same distance ($2$), opposite sides — exactly what a reflection guarantees. If your image is the wrong distance from the mirror, recheck which coordinate you negated.

🔑 Key Idea: Reflections preserve distance from every point to the mirror line, which is why the image is a true mirror copy — congruent to the original.

Reflect the Point 🧮

Apply each reflection and enter the new coordinates.

1) Reflect $(7, -4)$ over the $x$-axis. New $x = \,?$ and new $y = \,?$ 2) Reflect $(-3, 8)$ over the $y$-axis. New $x = \,?$ and new $y = \,?$

🔷 Translations and Reflections

Part 3 of 5 — Reflecting Over the Line $y = x$

🔑 The Idea: Besides the axes, a popular mirror line is the diagonal line $y = x$. Reflecting over it has a memorable rule: swap the coordinates.

Reflecting Over $y = x$

The line $y = x$ runs diagonally through the origin at $45^\circ$, passing through points like $(1,1)$, $(2,2)$, and $(-3,-3)$ — every point whose coordinates are equal.

To reflect over $y = x$, swap $x$ and $y$:

$(x,\ y) \;\longrightarrow\; (y,\ x)$

Example: reflect $A(2, 5)$ over $y = x$

Swap the coordinates:

$A(2,\ 5) \;\longrightarrow\; A'(5,\ 2)$

Example: reflect $B(-1, 4)$ over $y = x$

$B(-1,\ 4) \;\longrightarrow\; B'(4,\ -1)$

💡 No sign changes here — you only trade places. The new $x$ is the old $y$, and the new $y$ is the old $x$.

Match the Rule 🔽

Identify each image. Watch whether the move swaps or negates.

All Four Rules So Far

You now have the four reflection/translation rules Grade 8 needs. Keep this table handy.

⚠️ Don't confuse the rules! Reflecting over $y = x$ swaps coordinates (no negatives); reflecting over an axis negates one coordinate (no swapping).

Concept Check 🎯

One Last Check Before You Drill

When you reflect over $y = x$, a quick way to verify your answer: the original and image should be mirror images across the diagonal. If you connect a point to its image, that segment crosses $y = x$ at a right angle and is split in half by it.

For instance, $(2, 6)$ and its image $(6, 2)$ are symmetric across the diagonal — the midpoint $(4, 4)$ lies right on the line $y = x$. ✓

💡 If your swapped point and the original don't straddle the diagonal evenly, you probably negated a coordinate by accident. Remember: over $y = x$, swap only.

Reflect Over $y = x$ 🧮

Apply the swap rule and enter the new coordinates.

1) Reflect $(4, 9)$ over $y = x$. New $x = \,?$ and new $y = \,?$ 2) Reflect $(-6, 1)$ over $y = x$. New $x = \,?$ and new $y = \,?$

🔷 Translations and Reflections

Part 4 of 5 — Composing Two Transformations

🔑 Big Idea: A composition applies one transformation, then another, to the result. The order can matter — you must perform them in the stated sequence, feeding each image into the next rule.

How to Compose

To compose transformations, do the first one, write down the image, then apply the second rule to that image.

Example: reflect $A(3, 4)$ over the $x$-axis, then translate right $2$, up $1$

Step 1 — reflect over the $x$-axis $(x, y) \to (x, -y)$:

$A(3,\ 4) \;\longrightarrow\; A'(3,\ -4)$

Step 2 — translate $(x, y) \to (x + 2,\ y + 1)$, applied to $A'$:

$A'(3,\ -4) \;\longrightarrow\; A''(3 + 2,\ -4 + 1) = A''(5,\ -3)$

We write the final image as $A''$ (read "A double prime").

💡 Each transformation in a composition is still a rigid motion, so the final image is congruent to the original — the figure never changes size or shape, no matter how many flips and slides you chain together.

Compose Step by Step 🔽

Reflect $A(5, 2)$ over the $x$-axis, then translate left $3$, down $1$. Fill in each stage.

Order Can Matter

Swapping the order of two transformations can land the figure in a different spot, so always follow the sequence given.

Example: start with $P(2, 1)$

Path A — reflect over $y$-axis, then translate up $3$:

$P(2, 1) \to (-2, 1) \to (-2,\ 1 + 3) = (-2,\ 4)$

Path B — translate up $3$, then reflect over $y$-axis:

$P(2, 1) \to (2,\ 1 + 3) = (2, 4) \to (-2,\ 4)$

Here both paths happen to land at $(-2, 4)$ because a vertical slide and a $y$-axis flip don't interfere. But change the moves — say a horizontal slide and a $y$-axis flip — and the two orders give different answers.

⚠️ Always follow the order given. Do the first transformation completely, then apply the second to its image. Don't try to combine them in your head before computing.

Concept Check 🎯

A Two-Step Routine for Compositions

When a problem chains two transformations, run this routine every time:

1. Read the order. Which transformation happens first? Underline it.
2. Apply step 1 to the original point. Write down the intermediate image — call it $A'$.
3. Apply step 2 to $A'$ (never to the original). Write the final image $A''$.

⚠️ The classic mistake is applying both rules to the starting point. Always feed the image of step 1 into step 2.

Compose It 🧮

Apply both steps in order and enter the final coordinates.

1) Reflect $(4, 3)$ over the $y$-axis, then translate up $5$. Final $x = \,?$ and final $y = \,?$ 2) Translate $(-2, 6)$ down $4$, then reflect over the $x$-axis. Final $x = \,?$ and final $y = \,?$

🔷 Translations and Reflections

Part 5 of 5 — Congruence, Mixed Practice & Mastery Check

You can now (1) translate points and figures, (2) reflect over the $x$-axis, $y$-axis, and $y = x$, and (3) compose two moves in order. This last part ties it together with congruence and a final quiz.

Why Translations and Reflections Show Congruence

Two figures are congruent ($\cong$) when one can be mapped exactly onto the other using a sequence of rigid motions — translations, reflections, and rotations.

Because translations and reflections never change distances or angles:

• Corresponding side lengths stay equal.
• Corresponding angle measures stay equal.

So if you can slide and/or flip $\triangle ABC$ right on top of $\triangle DEF$, then $\triangle ABC \cong \triangle DEF$.

Example

$\triangle ABC$ has vertices $A(1,2)$, $B(4,2)$, $C(1,6)$. Reflect it over the $y$-axis:

$A(1,2) \to A'(-1,2), \quad B(4,2) \to B'(-4,2), \quad C(1,6) \to C'(-1,6)$

Side $AB$ has length $\lvert 4 - 1 \rvert = 3$; side $A'B'$ has length $\lvert -4 - (-1) \rvert = 3$. The lengths match — the reflection preserved them, confirming $\triangle ABC \cong \triangle A'B'C'$.

🔑 Key takeaway: Finding a translation/reflection that maps one figure onto another is exactly how Grade 8 proves the figures are congruent.

Name That Transformation 🔽

Each row shows a pre-image and image. Pick the single transformation that maps one to the other.

Quick Reference

Everything from this lesson in one place — use it for the mixed practice and exit quiz below.

⚠️ Reflecting over an axis negates one coordinate; reflecting over $y = x$ swaps them. Left/down are negative for translations. The image is always congruent to the original.

Mixed Practice 🎯

You're Ready

Nice work. You can now slide figures with a translation rule, flip them over the $x$-axis, $y$-axis, and the diagonal $y = x$, chain two moves into a composition, and explain why the result is always congruent to the original.

One final challenge below combines a translation and a reflection. Take it slow: do the translation first, write the intermediate point, then reflect.

🔑 Remember: read the order, work one step at a time, and watch your signs. Answer all three to finish the lesson.

Exit Quiz ✅

Answer all three to finish the lesson.