2D Shapes 3D Shapes Angles Area Circles Cones Coordinate Geometry Cylinders Dilation Distance Formula Equations Of Circles Euclidean Geometry Formulas For Perimeter Geometry Lines Measurement Midpoint Formula Planes Points Polygons Prisms Pyramids Quadrilaterals Reflection Rotation Spheres Surface Area Tangent Transformations Translation Triangles Volume

Translation in Mathematics

The "Slide" Transformation on the Coordinate Plane

In geometry, a Translation is a type of transformation that slides a figure from one position to another without turning, flipping, or resizing it.

[Image of translation geometry]

Imagine sliding a book across a table. The orientation of the book doesn't change; only its location does. This is the simplest form of rigid transformation.

1. How Translation Works

Every point of the object must be moved in the same direction and for the same distance. We describe this movement using a Translation Vector.

If you want to move a shape:

  • Horizontally: Add or subtract from the x-coordinate.
  • Vertically: Add or subtract from the y-coordinate.

2. The Coordinate Rule

To translate a point P(x, y) by a units horizontally and b units vertically, use the following formula:

(x, y) → (x + a, y + b)
  • If a is positive, move right. If negative, move left.
  • If b is positive, move up. If negative, move down.

3. Example 1: Basic Translation

Translate the point A(2, 3) by the vector <4, 1>. This means moving right 4 units and up 1 unit.

  • New x = 2 + 4 = 6
  • New y = 3 + 1 = 4

The new point is A'(6, 4).

4. Example 2: Handling Negatives

Translate the point B(-5, 2) by the rule: "Left 3, Down 4".

In math terms, "Left 3" means a = -3 and "Down 4" means b = -4.

New x = -5 + (-3) = -8
New y = 2 + (-4) = -2

The new point is B'(-8, -2).

5. Properties of Translation

Because translation is a Rigid Transformation (Isometry), several properties are always preserved:

  • Side Lengths: The size of the shape stays exactly the same.
  • Angle Measures: The corners of the shape do not change.
  • Orientation: The shape does not face a new direction (it isn't rotated).

Conclusion

Understanding Translation is fundamental to coordinate geometry. By applying simple addition or subtraction to coordinates, we can model movement in space, a concept widely used in computer graphics and navigation systems.