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Complete IGCSE Edexcel guide to performing and describing all four geometric transformations on coordinate grids. Step-by-step worked examples for grades 5-8.
The four geometric transformations are reflection (mirror flip across a line), rotation (turning around a fixed centre point by a specified angle), translation (sliding every point by the same column vector), and enlargement (resizing from a centre point by a scale factor). In the IGCSE exam, you must be able to both perform these on a grid and describe them in words with all required details.
Source: Edexcel IGCSE Mathematics (9-1) Specification 4MA1
A reflection flips a shape across a mirror line. Every point on the original shape is the same perpendicular distance from the mirror line as its image, but on the opposite side. The shape does not change size — only its orientation reverses.
| Mirror Line | Equation | Effect on (x, y) |
|---|---|---|
| x-axis | y = 0 | (x, -y) |
| y-axis | x = 0 | (-x, y) |
| y = x | y = x | (y, x) |
| y = -x | y = -x | (-y, -x) |
| Vertical line | x = k | (2k - x, y) |
To describe a reflection, you must state the equation of the mirror line. Writing "reflected in the y-axis" or "reflected in the line x = 3" are both acceptable as long as the line is correct.
A rotation turns a shape around a fixed point (the centre of rotation) by a specified angle in a specified direction (clockwise or anticlockwise). The shape stays the same size and the same orientation of its internal angles, but its position changes.
To find the centre of rotation, connect corresponding points on the original and image with straight lines, then construct perpendicular bisectors of those lines. The point where they meet is the centre.
To describe a rotation, you must state three things: the centre of rotation (as coordinates), the angle of rotation (90 degrees, 180 degrees, or 270 degrees for IGCSE), and the direction (clockwise or anticlockwise). Note that 180 degrees is the same in both directions, so direction is not required for 180-degree rotations.
A translation slides every point of a shape by the same distance in the same direction. It is described using a column vector where the top number gives horizontal movement (positive = right, negative = left) and the bottom number gives vertical movement (positive = up, negative = down).
To perform a translation, add the column vector to every vertex of the shape. To describe a translation, state the column vector. For example, "a translation by the vector (3, -2)" means every point moves 3 right and 2 down.
A translation is the only transformation that does not change the orientation of the shape — the image is identical in every way except position.
An enlargement changes the size of a shape from a fixed point called the centre of enlargement. Every distance from the centre to a vertex is multiplied by the scale factor. The image is mathematically similar to the original.
| Scale Factor | Effect |
|---|---|
| k > 1 | Image is larger, same side of centre |
| 0 < k < 1 | Image is smaller (reduction), same side of centre |
| k = -1 | Same size, opposite side (equivalent to 180-degree rotation about centre) |
| k < -1 | Image is larger and on opposite side of centre (inverted) |
To describe an enlargement, state the centre of enlargement (as coordinates) and the scale factor. To find the scale factor, divide the length of a side on the image by the corresponding side on the original. To find the centre, draw lines through corresponding vertices — they all meet at the centre of enlargement.
| Transformation | You Must State |
|---|---|
| Reflection | Mirror line (as an equation) |
| Rotation | Centre, angle, direction (CW or ACW) |
| Translation | Column vector |
| Enlargement | Centre, scale factor |
Start every description by naming the transformation type. Then give the required details. If you name the wrong type, you score zero regardless of the details you provide.
Triangle A has vertices at (1, 2), (3, 2), and (1, 5). It is reflected in the line y = x. Find the coordinates of the image triangle B.
Step 1: For a reflection in y = x, swap the x and y coordinates of each vertex.
Step 2: (1, 2) becomes (2, 1)
Step 3: (3, 2) becomes (2, 3)
Step 4: (1, 5) becomes (5, 1)
Answer: Triangle B has vertices (2, 1), (2, 3), and (5, 1)
Shape P is rotated 90 degrees clockwise about the origin to give shape Q. Shape Q is then translated by the vector (2, -3) to give shape R. Describe fully the single transformation that maps P to R.
Step 1: Track a test point, say (1, 0). After 90 degrees CW about (0, 0): (1, 0) maps to (0, -1).
Step 2: After translation by (2, -3): (0, -1) maps to (2, -4).
Step 3: Track a second point, say (0, 1). After 90 degrees CW about (0, 0): (0, 1) maps to (1, 0). After translation: (1, 0) maps to (3, -3).
Step 4: The combined transformation is a rotation. To find the centre and angle, use the perpendicular bisector method on the two pairs of original-image points.
Step 5: The single transformation is a rotation of 90 degrees clockwise about (2.5, -0.5) — though on IGCSE papers the centre will typically be integer coordinates. (Combined transformations at Standard tier usually produce a single transformation of the same type.)
Answer: Rotation, 90 degrees clockwise, about the centre found from perpendicular bisectors
Triangle T has vertices at (2, 3), (6, 3), and (4, 7). It is enlarged by scale factor -2 with centre of enlargement (3, 1). Find the vertices of the image.
Step 1: For each vertex, find the vector from the centre (3, 1) to the vertex, then multiply by -2.
Step 2: Vertex (2, 3): vector from centre = (-1, 2). Multiply by -2: (2, -4). Image = (3 + 2, 1 + (-4)) = (5, -3).
Step 3: Vertex (6, 3): vector from centre = (3, 2). Multiply by -2: (-6, -4). Image = (3 + (-6), 1 + (-4)) = (-3, -3).
Step 4: Vertex (4, 7): vector from centre = (1, 6). Multiply by -2: (-2, -12). Image = (3 + (-2), 1 + (-12)) = (1, -11).
Answer: Image vertices are (5, -3), (-3, -3), and (1, -11)
Incomplete descriptions: Not stating the mirror line for reflections, forgetting the direction for rotations, or omitting the centre for enlargements. Each missing piece costs marks.
Mixing up clockwise and anticlockwise: 90 degrees clockwise is the same as 270 degrees anticlockwise. If the question asks for the direction, be consistent with the smallest angle.
Drawing enlargements on the wrong side: With a negative scale factor, the image is on the opposite side of the centre. Many students place it on the same side and just make it bigger.
Reflecting in the wrong axis: Check carefully whether the mirror line is x = k or y = k. Confusing horizontal and vertical lines is surprisingly common under exam pressure.
Use tracing paper: You are allowed tracing paper in the IGCSE exam. Trace the original shape, then physically rotate or reflect to check your answer.
Always name the transformation first: Before giving details, write the type. If you call it the wrong type, you cannot recover marks from the details.
Check with one vertex: After performing a transformation, verify that one vertex of the image is correct before plotting the rest. This catches errors early.
Know the quick checks: Same size but flipped = reflection. Same size but turned = rotation. Same size, same orientation, different position = translation. Different size = enlargement.
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