🎯⭐ INTERACTIVE LESSON

Translation

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Translation - Complete Interactive Lesson

Part 1: 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

Section
What a Translation Really Does
Vector Notation and Coordinate Rules
Translating a Whole Figure
Composing Two Translations
Common Mistakes and How to Avoid Them

🔑 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 $\vec{v} = \langle a,\, b \rangle$ :

T_{\vec{v}} (x, y)

✏️ 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 $(x, y) \to (x$ .

🔁 Composing Two Translations

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

Check: Reading and Composing Translations

⚠️ Common Mistakes

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

Exit Quiz: Translation Mastery

T_{v} (x, y) = (x + a, y + b)

Notation	What it means
$\langle a,\, b \rangle$	Move $a$ units in the $x$ -direction and $b$ units in the $y$ -direction
$(x, y) \to (x + a,\, y + b)$	Coordinate rule for the same translation
$a > 0$	Shift to the right
$a < 0$	Shift to the left
$b > 0$	Shift up
$b < 0$	Shift down

💡 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

T_{\vec{v}}(P) - T_{\vec{v}}(Q) \;=\; (P + \vec{v}) - (Q + \vec{v}) \;=\; P - Q.

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

(x, y) \to (x - 3, y + 1)

Vertex	Rule applied	Image
$A(1, 2)$	$(1 - 3,\, 2 + 1)$	$A'(-2,\, 3)$
$B(4, 2)$	$(4 - 3,\, 2 + 1)$	$B'(1,\, 3)$
$C(2, 5)$	$(2 - 3,\, 5 + 1)$	$C'(-1,\, 6)$

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)

\vec{v}_2 = \langle a_2,\, b_2 \rangle

\vec{v}_1 + \vec{v}_2 \;=\; \langle a_1 + a_2,\; b_1 + b_2 \rangle.

First translation	Then translation	Net effect
$\langle 3,\, -2 \rangle$	$\langle 5,\, 7 \rangle$	$\langle 8,\, 5 \rangle$
$\langle -4,\, 1 \rangle$	$\langle 4,\, -1 \rangle$	$\langle 0,\, 0 \rangle$ (identity!)
$\langle 0,\, 6 \rangle$	$\langle -3,\, 0 \rangle$	$\langle -3,\, 6 \rangle$

🔑 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.)

\langle a,\, b \rangle

\langle 3,\, 4 \rangle

\langle 4,\, 3 \rangle

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.