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Welcome to GCSE Edexcel Maths revision.

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Topic M 26: Similarity and proportional measures.

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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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Similar figures have equal corresponding angles and proportional corresponding lengths.

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Congruent figures are the same size and shape, so their length scale factor is 1.

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Identify corresponding sides using matching angles and position, rather than choosing sides simply because they look similar on the diagram.

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Length scale factor from a smaller figure to a larger figure is the fraction with numerator open bracket corresponding large length close bracket and denominator open bracket corresponding small length close bracket ,

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end fraction .

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All corresponding lengths use the same factor.

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Two similar right-angled triangles

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Worked example: A side of 4 centimetres corresponds to 10 centimetres, so scale factor is 2.5.

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A corresponding 6 centimetres side becomes 15 centimetres; reversing the enlargement uses factor 0.4.

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For similar triangles, equal angles establish the same shape open bracket A A similarity close bracket .

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This is not a congruence test: equal angles alone do not force equal side lengths.

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Trigonometric ratios stay constant in similar right-angled triangles because corresponding sides scale together.

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This is why sine theta , cosine theta and tangent theta depend on the angle.

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Perimeters scale by the length factor.

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If each length triples, a perimeter of 18 centimetres becomes 54 centimetres.

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When the length factor is k, the area factor is k squared and the volume factor for similar solids is k cubed .

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Different dimensions require different powers.

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Worked example: Similar shapes with length ratio 2 to 5 have area ratio 4 : 25.

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An area of 12 square centimetres on the smaller shape corresponds to 12 multiplied by 25 over 4 equals 75 square centimetres .

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Worked example: Similar solids have length factor 3.

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Their volumes have factor 27, so a volume of 20 cubic centimetres becomes 540 cubic centimetres .

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To recover the length factor from an area factor, take a square root.

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An area factor 49 gives a length factor 7.

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From a volume factor 64, take a cube root to get 4.

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If similar containers have volume ratio 8 to 27, their height ratio is 2 to 3 and their surface-area ratio is 4 : 9.

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Mass scales like volume only if density is unchanged.

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Paint needed for a surface usually scales like area, whereas material filling a solid scales like volume.

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Similarity must be established before applying square or cube scale rules.

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Changing just one dimension of a cuboid does not create a similar solid.

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That completes Similarity and proportional measures.

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Revisit the notes and test yourself on the revision website.