Translation

TL;DR

Translation is moving every point of a shape in the same direction and by the same distance without changing its size or orientation. You can think of it as "sliding" a shape from one place to another. We use a translation vector to describe this movement, which tells us how much to move horizontally and vertically.

1. The Mental Model

Imagine you're sliding a book across a table. The book itself doesn't change size, spin, or flip over; it just moves from one spot to another. That's exactly what a translation does to a shape in geometry.

2. The Core Material

When we translate a shape, every single point on that shape moves the exact same way. This means the shape keeps its original size, shape, and orientation (it doesn't rotate or reflect).

Translation Vector

Photo by Steve A Johnson on Pexels

The key to a translation is the translation vector. This is usually written as a column vector, like this:

$\begin{pmatrix} x \\ y \end{pmatrix}$

• The top number ($x$) tells you how many units to move horizontally.
  • A positive $x$ means move right.
  • A negative $x$ means move left.
• The bottom number ($y$) tells you how many units to move vertically.
  • A positive $y$ means move up.
  • A negative $y$ means move down.

How to Apply a Translation

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To translate a point, you simply add the translation vector to its coordinates.

If you have a point $P = (a, b)$ and a translation vector $T = \begin{pmatrix} x \\ y \end{pmatrix}$, the new point, $P'$, after translation will be:

$P' = (a+x, b+y)$

When translating an entire shape, you apply this same rule to every vertex (corner) of the shape. Once all the vertices are translated, you connect the new vertices to form the translated shape.

Here's how you can visualize the process:

graph TD
    A["Start with Original Shape/Point"] --> B["Identify Coordinates (e.g., (2,3))"]
    B --> C["Determine Translation Vector (e.g., (4,-1))"]
    C --> D["Add Vector Components to Coordinates (2+4, 3-1)"]
    D --> E["Plot New Translated Point/Shape (e.g., (6,2))"]
    E --> F["Result: Translated Shape/Point"]

3. Worked Example

Let's translate a triangle with vertices $A(1, 2)$, $B(3, 1)$, and $C(2, 4)$ using the translation vector $T = \begin{pmatrix} 3 \\ -2 \end{pmatrix}$.

1. Translate Point A:
   $A(1, 2)$ becomes $A'(1+3, 2+(-2)) = A'(4, 0)$

2. Translate Point B:
   $B(3, 1)$ becomes $B'(3+3, 1+(-2)) = B'(6, -1)$

3. Translate Point C:
   $C(2, 4)$ becomes $C'(2+3, 4+(-2)) = C'(5, 2)$

So, the new triangle will have vertices at $A'(4, 0)$, $B'(6, -1)$, and $C'(5, 2)$. If you were to plot these, you'd see the entire triangle has slid 3 units to the right and 2 units down.

4. Key Takeaways

• A translation moves every point of a shape by the same distance and in the same direction.
• The shape's size, orientation, and angles remain unchanged during a translation.
• A translation vector $\begin{pmatrix} x \\ y \end{pmatrix}$ dictates the horizontal ($x$) and vertical ($y$) shift.
• Positive $x$ means right, negative $x$ means left. Positive $y$ means up, negative $y$ means down.
• To translate a point $(a,b)$ by vector $\begin{pmatrix} x \\ y \end{pmatrix}$, the new point is $(a+x, b+y)$.
• For shapes, apply the translation to each vertex individually.

Common Mistakes to Avoid:
* Mixing up x and y: Always remember the top number is for horizontal movement (x-axis) and the bottom is for vertical (y-axis).
* Forgetting signs: A negative number means moving left or down, not just "move."
* Changing the shape: A translation should never make the shape bigger, smaller, or rotate it. If it does, you've done something wrong.
* Only moving one point: Remember to move all vertices of a shape, not just one.

5. Now Try It

Draw a square with vertices at $P(0, 0)$, $Q(2, 0)$, $R(2, 2)$, and $S(0, 2)$ on a coordinate plane. Then, apply a translation using the vector $\begin{pmatrix} -3 \\ 1 \end{pmatrix}$. Plot the new vertices $P'$, $Q'$, $R'$, $S'$ and draw the translated square. What are the coordinates of the new vertices?