Geometry & measures
Transformations and Constructions
Year 10 · Year 11
- ✓By the end of this lesson students will be able to perform and describe reflections, rotations, translations, and enlargements.
- ✓By the end of this lesson students will be able to identify the properties and invariant points of each transformation.
- ✓By the end of this lesson students will be able to construct perpendicular bisectors, angle bisectors, and specific triangles using a compass and straight edge.
- ✓By the end of this lesson students will be able to identify and draw loci of points satisfying given conditions.
- ✓By the end of this lesson students will be able to solve problems involving combinations of transformations and loci.
Key concepts
A reflection creates a mirror image of a shape across a line, called the mirror line. Every point in the original shape is the same perpendicular distance from the mirror line as the corresponding point in the reflected image. The orientation of the shape changes. Reflections are congruent transformations.
A rotation turns a shape around a fixed point, called the centre of rotation, by a specified angle and direction (clockwise or anticlockwise). The shape's size and orientation remain the same, but its position changes. Rotations are congruent transformations.
A translation moves a shape from one position to another without changing its size, shape, or orientation. It is described by a column vector, where the top number indicates horizontal movement (positive for right, negative for left) and the bottom number indicates vertical movement (positive for up, negative for down). Translations are congruent transformations.
An enlargement changes the size of a shape by a given scale factor from a fixed point, called the centre of enlargement. The shape's orientation remains the same unless the scale factor is negative, in which case the image is inverted and on the opposite side of the centre. Enlargements are similar transformations (shape is preserved, size changes).
Geometric constructions involve drawing accurate figures using only a compass and a straight edge (ruler without markings). Key constructions include perpendicular bisectors, angle bisectors, and specific triangles.
A locus is the set of all points that satisfy a given geometric condition. Common loci include: points equidistant from a point (a circle), points equidistant from a line (a pair of parallel lines), points equidistant from two points (a perpendicular bisector), and points equidistant from two intersecting lines (an angle bisector).
Key facts to remember
- 1Reflections, rotations, and translations are congruent transformations (preserve size and shape).
- 2Enlargements are similar transformations (preserve shape, change size).
- 3To describe a reflection, you need the mirror line.
- 4To describe a rotation, you need the centre of rotation, the angle, and the direction (clockwise/anticlockwise).
- 5To describe a translation, you need the translation vector.
- 6To describe an enlargement, you need the centre of enlargement and the scale factor.
- 7A negative scale factor in an enlargement means the image is inverted and on the opposite side of the centre.
- 8Loci are sets of points satisfying specific conditions, often constructed using a compass and straight edge.
Worked examples
Example 1
Triangle A has vertices at (2, 1), (5, 1) and (2, 3). Reflect triangle A in the line y = x. Label the image B.
Answer
The vertices of triangle B are (1, 2), (1, 5) and (3, 2).
Remember that for reflection in the line y=x, the rule is (x, y) → (y, x).
Example 2
Triangle C has vertices at (1, 2), (3, 2) and (1, 4). Rotate triangle C 90° anticlockwise about the point (0, 0). Label the image D.
Answer
The vertices of triangle D are (-2, 1), (-2, 3) and (-4, 1).
If the centre of rotation is not the origin, translate the shape so the centre is at the origin, perform the rotation, then translate back.
Example 3
Describe fully the single transformation that maps triangle E with vertices (1, 1), (3, 1), (1, 2) onto triangle F with vertices (5, 3), (9, 3), (5, 5).
Answer
Enlargement, scale factor 2, centre (-3, -1).
Always state all three pieces of information for an enlargement: type, scale factor, and centre.
Example 4
Construct the perpendicular bisector of the line segment AB, where A is at (1, 2) and B is at (7, 2). Describe the locus of points equidistant from A and B.
Answer
The perpendicular bisector of AB is the line x = 4. The locus of points equidistant from A and B is the perpendicular bisector of the line segment AB.
Ensure your construction arcs are clearly visible as they are part of your working.
Common mistakes
- ✗Forgetting to state all necessary information when describing a transformation (e.g., omitting the centre of rotation or enlargement).
- ✗Incorrectly identifying the direction of rotation (clockwise vs. anticlockwise).
- ✗Applying the wrong rule for reflections in specific lines (e.g., confusing y=x with y=-x).
- ✗Not showing construction arcs when performing constructions, which are essential for marks.
- ✗Confusing the perpendicular bisector of a line segment with the angle bisector of two lines.
- ✗Misinterpreting the meaning of 'locus' in a problem, leading to an incorrect construction or description.
Exam tips
- ★Always use a sharp pencil, ruler, and compass for all transformations and constructions. Accuracy is key.
- ★When describing a transformation, be precise and include all required parameters (e.g., 'Rotation, 90° anticlockwise, centre (0,0)').
- ★For constructions, leave all construction lines and arcs clearly visible as they demonstrate your method.
- ★Practice identifying the type of transformation from an object and its image, and then finding its parameters.
- ★For loci questions, carefully read the condition and identify which standard construction (e.g., perpendicular bisector, angle bisector, circle) is required.
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