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C4.4 Similarity

Syllabus
0580–2028–2029
Topic
C4.4
Level
Core

Calculate corresponding lengths in similar shapes

Similar shapes have the same shape: corresponding angles are equal and every corresponding length is multiplied by one common scale factor.

k=new corresponding lengthoriginal corresponding length,new length=k×original lengthk=\frac{\text{new corresponding length}}{\text{original corresponding length}},\qquad \text{new length}=k\times\text{original length}

Match sides before calculating. In △ABC∼△PQR\triangle ABC\sim\triangle PQR, the stated order gives A↔PA\leftrightarrow P, B↔QB\leftrightarrow Q and C↔RC\leftrightarrow R, so ABAB corresponds to PQPQ, BCBC to QRQR and ACAC to PRPR.

Choose one complete pair of corresponding lengths and divide in the direction you need to find kk. Apply that same multiplier to the matching unknown side. Keep every ratio in the same shape order; an equivalent proportion is also valid.

Two similar plant pots have corresponding diameters 2727 cm and 3333 cm. If the smaller height is 21.621.6 cm, the enlargement factor is 33/2733/27. The larger height is 21.6×33/27=26.421.6\times33/27=26.4 cm.

An enlargement factor greater than 11 must produce a longer matching side; a reduction factor between 00 and 11 must produce a shorter one. This size check often catches an inverted ratio.

Do not match sides by their page position or apparent length: a shape may be rotated or reflected, and diagrams may not be to scale. Use vertex order, equal-angle information or the roles of the sides.