If the similarity ratio (the ratio comparing the sizes of two figures) is known, the lengths of corresponding sides can be found using a proportion.
How to Use Similarity Ratios
- If the similarity ratio is m : n, the ratio of corresponding sides is also m : n
- For the proportion a : b = c : x, calculate x = b × c ÷ a (the product of the means equals the product of the extremes)
Example 1: There are two similar triangles with a similarity ratio of 3 : 4. If one side of the smaller triangle is 9 cm, what is the length of the corresponding side of the larger triangle?
Solution:
① Set up a proportion → 3 : 4 = 9 : x
② x = 9 × 4 ÷ 3 = 12 cm
Solution:
① Set up a proportion → 3 : 4 = 9 : x
② x = 9 × 4 ÷ 3 = 12 cm
Example 2: Point D is taken on side BC of △ABC, and EF is drawn parallel to AD (E is on AB and F is on AC). If BE = 6 and EA = 4, find BD : DC.
Solution: Since there are parallel lines, BD : DC = BE : EA = 6 : 4 = 3 : 2
Solution: Since there are parallel lines, BD : DC = BE : EA = 6 : 4 = 3 : 2
The rule for “parallel lines and ratios of sides” (including the midpoint theorem) is closely connected to similarity.