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12.2 Solving transformation problems

12.2 Solving transformation problems

You already know that a shape can be transformed by a re#ection, rotation or translation.
When a shape undergoes any of these three transformations it only changes its position. Its shape and
size stay the same. Under these three transformations, an object and its image are always congruent.
When you re#ect a shape on a coordinate grid you need to know the x = –1 y x = 2

equation of the mirror line. 3
2
All vertical lines are parallel to the y-axis and have the equation x = ‘a number’. 1 y = 1
All horizontal lines are parallel to the x-axis and have the equation y = ‘a number’. 0 x
Some examples are shown on the grid on the right. –3 –2 –1 –1 1 2 3
–2 y = –2
When you rotate a shape on a coordinate grid you need to know the –3
coordinates of the centre of rotation, and the size and direction of the turn.
When you translate a shape on a coordinate grid, you can describe its movement with a column vector.
4  

$is is an example of a column vector:  
5  
$ e top number states how many units to move the shape right (positive number) or le% (negative number).

$e bottom number states how many units to move the shape up (positive number) or down
(negative number).
4   If the scale on the grid
For example:   means ‘move the shape 4 units right and 5 units up’
5   is one square to one
unit, the numbers tell
−  2 
  means ‘move the shape 2 units le% and 3 units down’. you how many squares
−  3  to move the object up
You can use any of these three transformations to solve all sorts of problem. or across.

Worked example 12.2a
The diagram shows a triangle on a coordinate grid. y
Draw the image of the triangle after each of these translations. 3
3    2  −  3   −  1 2
a   b   c   d  
2   1 −    1  −  3  1
0 x
–4 –3 –2 –1 1 2 3 4
–1
–2
–3

y a Move the original triangle (object) 3 squares right and 2 squares up.
3 b Move the original triangle 2 squares right and 1 square down.
a
2 c Move the original triangle 3 squares left and 1 square up.
c d Move the original triangle 1 square left and 3 squares down.
1
0 x
–4 –3 –2 –1 1 b 2 3 4
–1
–2
d
–3

12 Tessellations, transformations and loci 113