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10.1 Translation

GCSE Maths Foundation · Edexcel · Chapter 10: Transformations

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: sliding shapes using column vectors, and describing a translation. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Understand that a translation slides a shape without turning or resizing it
  • Read and write column vectors
  • Translate a point or a shape by a column vector
  • Describe a translation using a column vector

Key words

  • Transformation: A change to the position or size of a shape: translation, reflection, rotation or enlargement.
  • Translation: A slide. Every point of the shape moves the same distance in the same direction.
  • Object, image: The object is the shape you start with. The image is the shape after the transformation.
  • Column vector: Two numbers written one above the other. Top: move right (+) or left (−). Bottom: move up (+) or down (−).
  • Congruent: Exactly the same shape and size.

What a translation does

A translation slides a shape. The image is congruent to the object, and it faces the same way: it is not turned or flipped. We describe the slide with a column vector such as .

  • ; Meaning 3 right; (2, 4) moves to (5, 4)
  • ; Meaning 2 down; (2, 4) moves to (2, 2)
  • ; Meaning 4 left, 1 up; (2, 4) moves to (−2, 5)
  • ; Meaning 1 left, 5 down; (2, 4) moves to (1, −1)

Key fact: To translate a point, add the top number to the x-coordinate and the bottom number to the y-coordinate.

Translating a shape

Move each vertex (corner) by the vector, then join up the new vertices. Count squares carefully, or add to the coordinates. Label the image, for example T′.

Worked example: translating a triangle

Triangle T has vertices (1, 1), (3, 1) and (1, 4). Translate T by .
(1, 1) → (1 − 4, 1 + 2) = (−3, 3). (3, 1) → (−1, 3). (1, 4) → (−3, 6).

Answer: T′ has vertices (−3, 3), (−1, 3) and (−3, 6)

Describing a translation

Choose one vertex of the object and find the same vertex on the image. Work out how far it has moved across and up.

Key fact: Vector = image − object: (x of image − x of object) on top, (y of image − y of object) underneath.

Worked example: finding the vector

A translation moves P(2, 1) to P′(−1, 5).
Across: −1 − 2 = −3 (3 left). Up: 5 − 1 = 4 (4 up).

Answer: translation by

To describe a translation fully you must say translation and give the column vector. Nothing else is needed.

Going back and combining

To undo a translation, move the same distance the opposite way: change the sign of both numbers. Two translations one after the other make a single translation: add the tops and add the bottoms.

Worked example: undoing and combining

The reverse of is .
followed by is the same as = .

Answer: and

Common mistakes

  • Writing the vector upside down. The x-move (left or right) always goes on top.
  • Mixing up signs: left and down are negative, right and up are positive.
  • Moving only one vertex, or counting from the wrong vertex. Every vertex moves by the same vector.
  • Working out object − image instead of image − object, which gives the reverse translation.

Revision questions

Try these, then check your answers below each one.

Question 1. Mia says, “A translation can make a shape bigger.” Is Mia correct?

Answer: No

Working: A translation slides every point the same distance in the same direction. The image is congruent to the object, so its size does not change. Mia is wrong.

Related lessons

Continue in the full course

This lesson is section 10.1 (EGCMF1001) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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