Congruency, Enlargement and Combined Transformations · 5.3.4
Problems on combined transformations
Students apply a sequence of two transformations to solve multi-step problems, finding images, objects, or areas, and also find one single transformation (translation, reflection, rotation or enlargement) that has exactly the same overall effect as the combined pair.
The official learning standard (5.3.4)
“Solve problems involving combined transformation, including describing a single transformation equivalent to a combined transformation.”
What it means
Students apply a sequence of two transformations to solve multi-step problems, finding images, objects, or areas, and also find one single transformation (translation, reflection, rotation or enlargement) that has exactly the same overall effect as the combined pair.
How it is examined
This is a common extended Paper 2 question: students perform a combined transformation on a shape on a grid, then are asked to state and fully describe the single transformation with the same effect, testing both computation and conceptual understanding.
Worked example
Point A has coordinates (0, 2). A is reflected in the line x = 1 to form A′, and then A′ is reflected in the line x = 4 to form A″.
Find the coordinates of A″, and describe the single transformation equivalent to this combined transformation.
- Reflect A(0, 2) in x = 1: A is 1 unit to the left of the line, so A′ is 1 unit to the right of it, giving A′(2, 2).
- Reflect A′(2, 2) in x = 4: A′ is 2 units to the left of the line, so A″ is 2 units to the right of it, giving A″(6, 2).
- A(0, 2) maps directly to A″(6, 2): this is a translation by the vector (6, 0).
- Check: two reflections in parallel lines x = a and x = b give a translation of 2(b − a) along the x-axis; here 2(4 − 1) = 6, which matches.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
What does 'a single transformation equivalent to a combined transformation' mean?
It is one transformation, a translation, reflection, rotation or enlargement, that maps the same object to the same final image as the two combined transformations performed one after another.
What single transformation results from two reflections in parallel lines?
A translation. Its direction is perpendicular to the two parallel lines, and its magnitude is twice the perpendicular distance between them.
What single transformation results from two reflections in intersecting lines?
A rotation about the point where the two lines intersect, through an angle equal to twice the angle between the two lines, in the direction from the first mirror line towards the second.