L1 · Scale factor and similar figures

Learning goal

“I can find the scale factor between two similar figures (image / original, same units), use it to find missing lengths, and decide whether two figures are similar by checking that every pair of corresponding lengths gives the same ratio.”

Similar — same shape, possibly different size: one figure can be obtained from the other by a scale factor (the curriculum also allows a rotation). Every corresponding length is multiplied by the same number.

Scale factor (k) — the multiplier from original to image: k = image length / corresponding original length, in the same units.

Corresponding lengths — lengths in the same relative position on the two figures: the width of the original matches the width of the image, and so on.

Enlargement / reduction — an enlargement has k > 1; a reduction has 0 < k < 1; k = 1 keeps the size (the figures are congruent).

Worked example

A photo 10 cm wide x 15 cm tall is enlarged to 14 cm wide x 21 cm tall. Find the scale factor and confirm the figures are similar. Step 1 (one pair): k = image / original = 14 / 10 = 1.4. Step 2 (check the other pair): 21 / 15 = 1.4. Same multiplier - the enlargement is honest. Step 3 (name it): k = 1.4 > 1, so this is an enlargement, and the two photos are similar. Step 4 (apply it): a detail 5 cm long in the original is 5 x 1.4 = 7 cm in the enlargement. Say it plainly: the scale factor is ONE number that must work for EVERY corresponding length. If two pairs give two different ratios, the figures are not similar - full stop.

Check yourself — explain it

Both side pairs had to be checked because...

Paired problem — example still visible

An original segment is 8 cm; its image is 12 cm. Find k and name the change. -> k = 12 / 8 = 1.5 - an enlargement.

Faded completion — last step is yours

"With k = 0.75, an original length of 20 cm becomes 20 x 0.75 = ______ cm." -> 15 cm (and k < 1, so it is a reduction).

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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