Mini Mathematics
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Congruence and similarity

Match corresponding parts before comparing lengths, angles and scale.

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Congruent figures have the same size and shape

Congruent figures match after a translation, rotation or reflection. Corresponding sides have equal lengths and corresponding angles are equal. A turned or reflected triangle is not a different size merely because its orientation changes.

Similar figures share shape at a common scale

Similar figures have equal corresponding angles and proportional corresponding sides. If corresponding sides are 4 cm and 10 cm, the enlargement factor from the smaller to the larger is 10/4 = 2.5. A second smaller side of 6 cm becomes 15 cm. From larger to smaller, use the reciprocal factor 0.4.

Match the vertices before writing the ratios. If triangle ABC corresponds to triangle PQR, then AB matches PQ, BC matches QR and AC matches PR. Writing AB/QR would compare unmatched sides unless the correspondence is different.

Check your understanding 1 Check more than one pair

Two rectangles have dimensions 4 cm by 6 cm and 6 cm by 9 cm. Are they similar? Compare a 4 cm by 6 cm rectangle with a 6 cm by 8 cm rectangle.

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The first pair are similar: both corresponding side ratios are 1.5. The second pair are not: 6/4 = 1.5 but 8/6 is not 1.5. Both have right angles, but their side proportions differ.

A single side ratio is not enough

A photograph is resized from 8 cm by 12 cm to 12 cm by 18 cm. Explain why the image’s shape is preserved. What happens if the new height is 16 cm instead?

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Both original dimensions were multiplied by 1.5 in the first resize. The second uses factors 1.5 and 4/3, so it stretches the directions differently and does not preserve similarity.

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