Geometric Transformations: Reflection, Rotation, Enlargement and Translation
The four transformations are reflection, rotation, translation and enlargement, and each has a set of details that must be stated for a description to earn full marks. A reflection needs the equation of the mirror line. A rotation needs the angle, the direction (clockwise or anticlockwise) and the centre of rotation. A translation needs a column vector. An enlargement needs a scale factor and a centre. A fractional scale factor makes the image smaller, and a negative scale factor puts the image on the opposite side of the centre and turns it upside down. Only enlargement changes size; the other three produce a congruent image.
Before you start
No specific prerequisites - this is a good place to start.
Method
- Identify the type first from what has changed: same size and orientation means a translation, same size and flipped means a reflection, same size and turned means a rotation, and different size means an enlargement.
- For a reflection, find the mirror line by joining a point to its image and finding the perpendicular bisector, then state the line as an equation such as x = 3, y = -1 or y = x.
- For a rotation, find the centre by joining two points to their images and constructing the perpendicular bisectors of both; where they cross is the centre. State the angle and direction as well.
- For a translation, count the horizontal and vertical movement from one point to its image and write it as a column vector, with right and up positive.
- For an enlargement, find the scale factor by dividing an image length by the corresponding original length, then find the centre by drawing rays through pairs of corresponding points and seeing where they meet.
- Describe a transformation with ONE transformation only unless the question asks for a combination, and give every required detail: a description missing the centre or the direction loses the mark even when the transformation is right.
Worked example
Triangle A has vertices (1, 1), (3, 1) and (1, 4). It is enlarged by scale factor 2 with centre (0, 0) to give triangle B. Find the vertices of B, and state what happens to the area.
- For an enlargement centred on the origin, multiply both coordinates of each point by the scale factor.
- The vertex (1, 1) maps to (2, 2).
- The vertex (3, 1) maps to (6, 2).
- The vertex (1, 4) maps to (2, 8).
- For the area, lengths are multiplied by 2, so the area is multiplied by 2 squared = 4.
- Check with the actual areas: triangle A has base 2 and height 3, giving an area of 3, while triangle B has base 4 and height 6, giving an area of 12, which is 4 times as large as predicted.
Practice questions
Try each question, then tap to reveal the answer.
Q1What three details must be given to describe a rotation fully?Show answer
Answer: The angle, the direction (clockwise or anticlockwise) and the centre of rotation.
Q2Reflect the point (4, 3) in the line y = x.Show answer
Answer: (3, 4), because reflecting in y = x swaps the coordinates.
Q3Translate the point (2, -1) by the column vector 3 over 5.Show answer
Answer: (5, 4).
Q4What does a negative scale factor do in an enlargement?Show answer
Answer: It puts the image on the opposite side of the centre of enlargement and turns it upside down, as well as scaling it.
Q5Which of the four transformations does not produce a congruent image?Show answer
Answer: Enlargement, because it changes the size unless the scale factor is 1 or -1.
Q6An enlargement has scale factor one third. Is the image larger or smaller?Show answer
Answer: Smaller, one third of the original lengths.
Q7How do you find the centre of an enlargement from a diagram?Show answer
Answer: Draw straight rays through each pair of corresponding points and extend them; they all meet at the centre.
Exam-style questions
Written in the style of a IGCSE Maths exam paper, with a full mark scheme.
Triangle P has vertices (2, 1), (4, 1) and (2, 5). Triangle Q has vertices (2, -1), (4, -1) and (2, -5). Describe fully the single transformation that maps P onto Q.
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Shape S is enlarged by scale factor -2 with centre (1, 1). The point (3, 2) lies on S. Find the image of this point, and explain the effect of the negative scale factor.
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Free printable worksheet
Want more practice on paper? Download the geometric transformations: reflection, rotation, enlargement and translation worksheet pack - 6 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
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