Transformations Worked Examples for IGCSE Maths

Question

Triangle A has vertices (1,1), (3,1), (1,4). Triangle B has vertices (-1,1), (-3,1), (-1,4). Describe the single transformation that maps A to B.

Solution

1. 1

   Observe how points have changed

   (1,1) maps to (-1,1): x has been negated, y is unchanged. Same for all points.

   When x changes sign but y stays the same, it is a reflection in the y-axis.

2. 2

   Write the full description

   Reflection in the y-axis (the line x = 0)

   Every reflection description must include the mirror line.

Question

Rotate triangle with vertices A(1,1), B(4,1), C(4,3) by 90 degrees anticlockwise about the origin.

Solution

1. 1

   Apply the rotation rule

   For 90 degrees anticlockwise about (0,0): (x,y) maps to (-y,x)

   This is the standard rule for 90 degree anticlockwise rotation about the origin.

2. 2

   Transform each vertex

   A(1,1) -> (-1,1). B(4,1) -> (-1,4). C(4,3) -> (-3,4).

   Apply the rule to each vertex.

Question

Triangle P has vertices (1,1), (3,1), (1,2). Triangle Q has vertices (3,1), (9,1), (3,4). Describe the single transformation mapping P to Q.

Solution

1. 1

   Find the scale factor

   P has base 2 (from x=1 to x=3), Q has base 6 (from x=3 to x=9). Scale factor = 6/2 = 3.

   Compare corresponding side lengths.

2. 2

   Find the centre of enlargement

   Draw lines through corresponding vertices. Line through (1,1) and (3,1) extends from a centre. Line through (1,2) and (3,4): gradient = (4-2)/(3-1) = 1, so y - 2 = 1(x - 1), y = x + 1. At (1,1): y = 1+1 = 2, not 1. Try centre (0,1): from (0,1) to (1,1) is 1 unit right, 0 up. Times 3: 3 right, 0 up = (3,1). Correct. From (0,1) to (1,2) is 1 right, 1 up. Times 3: 3 right, 3 up = (3,4). Correct. Centre = (0,1).

   The centre of enlargement is the point from which all distances are multiplied by the scale factor.

3. 3

   Write the full description

   Enlargement, scale factor 3, centre (0, 1)

   An enlargement description must include both the scale factor and the centre.

Question

Translate triangle with vertices (2,3), (5,3), (2,6) by the vector (-3, 2).

Solution

1. 1

   Add the vector to each vertex

   (2,3) + (-3,2) = (-1,5). (5,3) + (-3,2) = (2,5). (2,6) + (-3,2) = (-1,8).

   Translation means adding the column vector to every point.