Ratios and Proportions

Scale and Similar Figures

Find a missing side of similar figures using proportional reasoning and cross-multiplication.

Example

Use a scale factor to find the missing corresponding side.

highlighted = computed this step

Step 1 — Corresponding sides

Set up the relationship.

69=8b′\frac{6}{9} = \frac{ 8 }{b'}
Match corresponding sides689b' ?x 3/2Corresponding side lengths use the same scale factor.

Step 2 — Find scale factor

Find the scale factor: 9 / 6 = 3/2.

9÷6=329 \div 6 = \hlmath{\frac{3}{2}}
Scale factor: 9 / 6 = 3/2689b' ?x 3/2Corresponding side lengths use the same scale factor.

Step 3 — Scale the matching side

Scale the matching side: 8 x 3/2 = 12.

8×32=128 \times \frac{3}{2} = \hl{12}
Scale the matching side: 8 x 3/2 = 12689b' = 12x 3/2Corresponding side lengths use the same scale factor.

Step 4 — Cross product check

Check the cross product: 9 x 8 = 72.

9×8=729 \times 8 = \hl{72}
Check: 9 x 8 = 72 and 6 x 12 = 72689b' = 12x 3/29 x 8 = 72; 6 x 12 = 72

Step 5 — Result

Read the final result.

b′=12b'= 12
Missing side b' = 12689b' = 12x 3/2Corresponding side lengths use the same scale factor.
scale-and-similar To find missing side b of similar figures where a/a' = b/b': - Cross-multiply: a'·b = a·b'. - Solve: b' = (a'·b) / a.