18.5 Similarity and similar lengths — Learn

Prerequisites: ratio, fractions, multiplication and division.

Objectives: identify similar shapes and find corresponding lengths using a common scale factor.

Similar shapes have equal corresponding angles and proportional corresponding lengths. The scale factor is the multiplier from one shape to the other.

Example. Similar triangles have corresponding sides $4$ cm and $10$ cm.

Scale factor from small to large is $10\div4$. $10\div4=2.5$. A side of $6$ cm on the smaller triangle becomes $6\times2.5=15$ cm.

To test if shapes are similar, compare ratios of corresponding lengths. All ratios must match.

Example. Compare side lengths $6,9,12$ with $10,15,20$.

$10\div6=\frac53$. $15\div9=\frac53$. $20\div12=\frac53$. The ratios match, so the side lengths are proportional.

Watch out. Keep corresponding sides in the same order. A ratio of small-to-large must not be compared with a ratio of large-to-small. Check that each multiplier matches.

Progress check

A mural is reproduced so that a tile edge of $8$ cm becomes $12$ cm. Work out the scale factor, then the matching length for an edge of $6$ cm on the original. Explain how the ratios confirm the scale.

You should now be able to

• explain what makes two shapes similar;
• calculate a scale factor using corresponding lengths;
• find missing lengths in similar shapes;
• test similarity by comparing corresponding ratios.