﻿ Unit 7 Section 2 : Translations

# Unit 7 Section 2 : Translations

Under a translation, every point is moved by the same amount in the same direction. If each point moves distance a in the x-direction and distance b in the y-direction, we use the 'vector' notation to describe this translation. For example, the translation described by the column vector is illustrated opposite; the translation moves the shape 6 units to the right and 2 units upwards.

Note that the actual shape does not change its orientation, only its position. It is not reflected or rotated.

## Example 1

What could each one of the following shapes be if it has 4 sides and:

(a)

Draw the square with corners at the points with coordinates (4, 0), (1, 3), (4, 6) and (7, 3).  (b)

The square is translated along the vector 52 . Draw the new square obtained by the translation.  For this translation each point should be moved 5 units to the right and 2 units up.

This diagram shows both squares and the vector that has been used to translate each corner.

## Example 2

The diagram below shows the shapes A, B, C and D. Along what vector would you translate:

 (a) D to A, (b) C to D, (c) A to B, (d) A to C ?  (a) D to A , 10 to the right and 3 up. (b) C to D , 1 to the right and 8 up. (c) A to B , 5 to the right and 10 down. (d) A to C , 11 to the left and 11 down.

## Exercises

Question 1

(a)

Draw the triangle which has corners at the points with coordinates (4, 1), (3, 5) and (1, 2).

(b)

Translate the triangle along the vector .

(c)

Write down the coordinates of the corners of the translated triangle.

, ,
Question 2

The following diagram shows the shape A which is translated to give the shapes B, C, D and E: Write down the vector that describes the translation from:

 (a) A to B, (b) A to C, (c) A to D, (d) A to E, (e) B to D.
Question 3

(a)

Join the points with coordinates (1, 1), (2, 3) and (5, 4) to form a triangle. Label this triangle A.

(b)

Translate the triangle A along the vector:

 (i) (ii) (iii) (iv)
Question 4

Write down the vector needed to translate the shape A to each of the other shapes shown on the following diagram: AB AC AD AE AF AG
Question 5

The point with coordinates (2, 3) is moved to the point with coordinates (7, 6) by a translation.

(a)

Describe the translation using a column vector.

(b)

Where would the point with coordinates (6, 1) move to under the same translation?

Question 6 The diagram shows three shapes, A, B and C:

Write down the vector for the translation that moves:

 (a) A to B, (b) B to C, (c) A to C.
There is relationship between these vectors: + = , i.e. for the top numbers (the x movement), 6 + 6 = 12 and for the bottom numbers (the y movement), 3 + (–4) = –1.
Question 7

The shape A has corners at the points with coordinates

 (4, 2) (4, –1) (6, –3) and (6, 0)
(a)

What is this shape?

(b)
A
B
C

The shape is translated along the vector to give shape B and then shape B is translated along the vector to give C.

Draw A, B and C.

(c)

What translation would take A straight to C ?

Question 8

(a)

Draw the triangle, A, that has corners at the points with coordinates (–7, –2), (–5, –5) and (–4, –2).

(b)

Translate this shape along the vector to obtain B.

(c)

Describe the translation that would take B to A.

Question 9 (a)

Draw three lines by joining the points with coordinates

 A: (4, 2) and (2, 4); B: (6, 4) and (6, 6); C: (2, 6) and (4, 8).
(b)

Describe how to translate each line to form a hexagon made up of the original and translated lines.

 AA': BB': CC':
Question 10

A parallelogram has corners at the points A, B, C and D. The points A, B and C have coordinates (1, 2), (2, 5) and (5, 3) respectively.

(a)

Draw the parallelogram.

(b)

State the coordinates of the fourth corner, D.

(c)

Describe the translation that moves A B onto D C.

(d)

Describe the translation that moves A D onto B C.