Similarity
Understand what makes two shapes similar, how to find scale factors, and how to use ratios of corresponding sides to find unknown lengths.
What is Similarity?
Two shapes are similar if they have exactly the same shape but may be different sizes. One is an enlargement or reduction of the other. Similar shapes have equal corresponding angles and proportional corresponding sides.
Properties of Similar Shapes
- ✓ Corresponding angles are equal
- ✓ Corresponding sides are in the same ratio
- ✓ The ratio is called the scale factor
Similar vs Congruent
- Congruent: Same shape AND same size
- Similar: Same shape, possibly different size
- All congruent shapes are similar (scale factor = 1)
Similar Triangles — Same Shape, Different Size
▵ABC
▵DEF
Scale factor = 12 ÷ 8 = 1.5. Each side of ▵DEF is 1.5 times the corresponding side of ▵ABC.
Scale Factor
The scale factor is the ratio by which all lengths are multiplied to go from one similar figure to another.
Scale Factor = Corresponding side in new figure ÷ Corresponding side in original figure
Scale factor > 1
🔍
Enlargement — the new shape is bigger
Scale factor = 1
✓
Congruent — same size
Scale factor < 1
🔎
Reduction — the new shape is smaller
Finding Unknown Sides
Because corresponding sides are in proportion, you can set up a ratio equation to find unknown lengths.
If ▵ABC ~ ▵DEF (similar), then:
AB/DE = BC/EF = AC/DF = scale factor
Use this to write a proportion and solve for the unknown side using cross-multiplication.
Key Vocabulary
Similar (~)
Shapes that have the same shape but different sizes. Denoted by the symbol ~.
Scale Factor
The ratio of corresponding sides between similar figures. Used to enlarge or reduce shapes.
Enlargement
A transformation that produces a similar figure that is larger. Scale factor is greater than 1.
Proportion
An equation stating that two ratios are equal. Used to find unknown sides in similar figures.
Worked Examples
▵ABC ~ ▵DEF. AB = 4 cm, BC = 6 cm, DE = 6 cm. Find EF.
Step 1: Scale factor = DE ÷ AB = 6 ÷ 4 = 1.5
Step 2: EF = BC × 1.5 = 6 × 1.5 = 9 cm
Answer: EF = 9 cm
A photograph 10 cm wide is enlarged to 25 cm wide. If the photo was 6 cm tall, how tall is the enlargement?
Step 1: Scale factor = 25 ÷ 10 = 2.5
Step 2: Height = 6 × 2.5 = 15 cm
Answer: 15 cm tall
Are these triangles similar? ▵PQR: angles 40°, 70°, 70°. ▵XYZ: angles 40°, 70°, 70°.
Step 1: Compare corresponding angles: 40° = 40°, 70° = 70°, 70° = 70°.
Step 2: All three pairs of angles are equal (AA similarity condition).
Answer: Yes, the triangles are similar (AA).
Knowledge Check
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Key Concepts Summary
- ●Similar shapes have the same shape but may be different sizes; denoted by ~.
- ●Corresponding angles are equal; corresponding sides are in the same ratio (scale factor).
- ●Scale factor = corresponding side of new figure ÷ corresponding side of original figure.
- ●Use ratios and proportions to find unknown sides in similar figures.
- ●Two triangles are similar if two pairs of angles are equal (AA condition).