Translation

Translation: Sliding Figures in the Plane

Focus: Move every point of a figure by the same vector — no turning, no flipping, no resizing.

Topics in This Lesson

🔑 Big Idea: A translation is the only rigid transformation that moves every point by the same vector. That single property is why translations preserve lengths, angle measures, and orientation — they are the gentlest isometry.

What You'll Master

• Read and apply translation rules in the form $(x, y) \to (x + a,\, y + b)$
• Convert between vector notation $\langle a,\, b \rangle$ and coordinate rules
• Translate any polygon by translating each vertex consistently
• Recognize when two consecutive translations can be combined into one

Entrance Quiz: Translation Readiness

🧭 Vector Notation and Coordinate Rules

A translation is described completely by a single translation vector $\vec{v} = \langle a,\, b \rangle$:

💡 Reading a translation backwards: if you know a starting point $P$ and its image $P'$, you can recover the vector by subtracting: $\vec{v} = P' - P$. For $P(2, -3)$ and $P'(5, 1)$, the translation vector is $\langle 3,\, 4 \rangle$.

Why translations preserve everything

For any two points $P$ and $Q$, the translated points satisfy

The vector between any two points is unchanged, so distances, angles, and orientation all stay exactly the same.

✏️ Translating a Whole Figure

To translate a polygon, translate each vertex with the same rule and connect the images in the same order.

Worked Example

Translate triangle $A(1, 2),\; B(4, 2),\; C(2, 5)$ by the rule $(x, y) \to (x - 3,\, y + 1)$.

Connect $A' \to B' \to C' \to A'$ in the same order as the original. Because every point shifted by $\langle -3,\, 1 \rangle$:

• $\overline{AB}$ and $\overline{A'B'}$ are the same length ($3$ units)
• $\angle A$ and $\angle A'$ have the same measure
• The triangle still goes counterclockwise (orientation preserved)

🔁 Composing Two Translations

If you translate by $\vec{v}_1 = \langle a_1,\, b_1 \rangle$ and then by $\vec{v}_2 = \langle a_2,\, b_2 \rangle$, the combined effect is a single translation by

🔑 Order doesn't matter for translations: $\vec{v}_1 + \vec{v}_2 = \vec{v}_2 + \vec{v}_1$. (This is unusual — for most combinations of rigid transformations, order does matter.)

Check: Reading and Composing Translations

⚠️ Common Mistakes

1. Confusing direction signs. Many students read "$x - 3$" as "move right 3" because they see the "3". Always trust the sign: a subtraction on the $x$-coordinate shifts left.

2. Applying the rule to only one vertex. Once you find one image point, it's tempting to "just count" to find the others on the graph. Counting from a graph introduces errors — use the coordinate rule for every vertex.

3. Swapping the role of $a$ and $b$. In $\langle a,\, b \rangle$, the first entry is always horizontal and the second is always vertical. Writing $\langle 3,\, 4 \rangle$ for "up 3, right 4" is wrong; the correct vector is $\langle 4,\, 3 \rangle$.

4. Forgetting that the translation vector is the same for every point. A translation does not depend on where the point is — it depends only on the vector.

Exit Quiz: Translation Mastery