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Welcome to GCSE Edexcel Maths revision.

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Topic M 29: Transformations and congruence.

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This video covers Higher tier.

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It includes the shared content and the labelled Higher extensions.

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A translation moves every point by the same vector. column vector, 3, minus 2 means 3 units right and 2 down.

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A complete description needs both components.

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A triangle translated three right and two down

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A reflection needs its mirror line, for example x equals 2 or y equals x.

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Perpendicular distances from corresponding points to the mirror line are equal.

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Reflecting open bracket a,

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b close bracket in the x-axis gives open bracket a,

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minus b close bracket ;

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in the y-axis gives open bracket minus a,

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b close bracket ;

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in y equals x gives open bracket b,

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a close bracket .

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Worked example: A triangle with vertices open bracket 1,

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1 close bracket ,

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open bracket 3,

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1 close bracket ,

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open bracket 1,

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3 close bracket reflects in the y-axis to open bracket minus 1,

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1 close bracket ,

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open bracket minus 3,

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1 close bracket ,

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open bracket minus 1,

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3 close bracket .

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Each corresponding point is equally far from the mirror line.

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Reflection of a triangle in the y-axis

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A rotation needs centre, angle and direction unless the angle is 180 degrees.

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A 90 degrees anticlockwise rotation about the origin sends open bracket a, b close bracket to open bracket minus b, a close bracket .

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Worked example: Rotating open bracket 3, 1 close bracket through 90 degrees anticlockwise about open bracket 0, 0 close bracket gives open bracket minus 1, 3 close bracket .

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Draw centre-to-point radii to check that the distance from the centre stays unchanged.

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A ninety-degree anticlockwise rotation

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An enlargement needs a centre and a scale factor.

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Draw rays from the centre through each vertex and multiply each distance from the centre by the scale factor.

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Worked example: Enlarging open bracket 3,

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1 close bracket by scale factor 2 about open bracket 1,

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1 close bracket doubles the relative vector open bracket 2,

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0 close bracket ,

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giving open bracket 5,

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1 close bracket .

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Doubling the original coordinates works only for a centre at the origin.

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Join each vertex and its image to the enlargement centre; corresponding points must lie on those rays.

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For scale factor 2, an original vertex at distance d moves to distance 2 d from the centre.

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An enlargement with scale factor two

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A positive scale factor below 1 reduces the shape.

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Enlargement by 1 over 2 halves every length but keeps all angles equal.

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Reflections, rotations and translations preserve lengths and angles, so images are congruent.

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Enlargement preserves angles and proportions; it generally changes size.

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Congruent triangles have corresponding sides and angles equal.

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Valid tests are S S S, S A S, A S A open bracket or A A S close bracket and R H S for right-angled triangles with equal hypotenuse and one other side.

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In S A S, the equal angle must be included between the two equal sides.

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S S A does not generally prove congruence; A A A proves similarity only.

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When using congruence in a proof, identify corresponding vertices in order, name the test, then use the corresponding equal lengths or angles.

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A negative enlargement places the image on the opposite side of the centre.

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Distances are multiplied by the magnitude of the factor; a factor minus 2 also reverses the direction of each centre-to-point vector.

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Worked example: Enlarging open bracket 3,

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1 close bracket by factor minus 2 about open bracket 1,

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1 close bracket gives open bracket minus 3,

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1 close bracket ,

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because open bracket 1,

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1 close bracket plus open bracket minus 2 close bracket multiplied by open bracket 2,

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0 close bracket equals open bracket minus 3,

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1 close bracket .

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For combined transformations, apply them in the stated order; changing the order can change the result.

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Two translations combine by adding their vectors.

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Two reflections in parallel lines produce a translation; two reflections in intersecting lines produce a rotation.

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Track a point and orientation to identify the resulting transformation.

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That completes Transformations and congruence.

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Revisit the notes and test yourself on the revision website.