Ratios and similar polygons
Similar figures have the same shape but not necessarily the same size. Ratios and proportions describe exactly how their measurements relate.
Ratios and proportions
A ratio compares two quantities, such as 3 : 4 or \(\frac{3}{4}\). A proportion states that two ratios are equal, \(\frac{a}{b} = \frac{c}{d}\), and is solved by cross-multiplying: \(ad = bc\). If \(\frac{x}{12} = \frac{5}{8}\), then 8x = 60 and x = 7.5.
Similar polygons
Two polygons are similar if their corresponding angles are congruent and their corresponding sides are proportional. The symbol ~ means 'is similar to'. As with congruence, the order of letters in a statement such as ABCD ~ EFGH shows which vertices correspond.
Scale factor
The ratio of corresponding side lengths is the scale factor. If a rectangle 4 cm by 6 cm is enlarged to 10 cm by 15 cm, the scale factor is 10/4 = 2.5. Both pairs of sides must give the same ratio for the figures to be similar: a 4 by 6 rectangle is not similar to a 5 by 6 one.
Checking similarity
Having equal angles is not enough for polygons in general: every rectangle has four right angles, but a square and a long thin rectangle are not similar. Having proportional sides is not enough either: a square and a non-square rhombus with equal sides have proportional sides but different angles. Polygons need both conditions.
Perimeters of similar figures
If two figures are similar with scale factor k, their perimeters are in the same ratio k. A triangle with perimeter 18 cm that is enlarged by a scale factor of 3 has a perimeter of 54 cm.
Scale drawings and maps
Architects' plans, maps, and models are similar to the real objects. A map scale of 1 : 50,000 means 1 cm on the map represents 50,000 cm, or 0.5 km. A distance of 7 cm on that map is 3.5 km on the ground.
The golden ratio
The golden ratio, about 1.618, appears when a rectangle's length-to-width ratio equals the ratio of their sum to the length. Removing a square from a golden rectangle leaves a smaller golden rectangle, similar to the original. Claims that it appears everywhere in art and nature are often exaggerated, though it does arise in some plant growth patterns.
Worked example
A model car is built at a scale of 1 : 24. The real car is 4.32 m long, so the model is 4.32 / 24 = 0.18 m, or 18 cm, long. If the model's wheels are 2.7 cm across, the real wheels are 2.7 x 24 = 64.8 cm across.
Key takeaways
A ratio compares quantities and a proportion sets two ratios equal, solved by cross-multiplying. Similar polygons have congruent corresponding angles and proportional corresponding sides; both conditions are needed. The scale factor is the ratio of corresponding sides, and perimeters scale by the same factor. Maps, plans, and models are similar figures defined by a scale. The golden ratio, about 1.618, describes rectangles that remain similar after a square is removed.
Flashcard
Comparison of two quantities
Check your understanding
A 4 cm by 6 cm rectangle is enlarged to 10 cm by 15 cm. What is the scale factor?