Similar Figures
Learn what makes figures similar and how to use proportions to find missing measurements.
Definition
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Worked Examples
Triangle ABC has sides 3 cm, 4 cm, and 5 cm. Triangle DEF has sides 6 cm, 8 cm, and 10 cm. Are these triangles similar?
Check if side ratios are equal
, , → All ratios equal 2
Verify the scale factor
Each side of DEF is exactly 2 times the corresponding side of ABC → Scale factor = 2
Conclude
Since all corresponding sides have the same ratio, the triangles are similar →
Answer: Yes, the triangles are similar with a scale factor of 2.
Common Mistakes
Confusing similar with congruent
Why it's wrong: Congruent figures are exactly the same size AND shape. Similar figures have the same shape but can be different sizes.
Correct: Similar = same shape, different size allowed. Congruent = same shape AND same size.
Not matching corresponding sides correctly
Why it's wrong: When comparing similar figures, you must match sides in the correct order based on their positions.
Correct: Always match the shortest side to shortest, longest to longest, or use angle positions to identify corresponding sides.
Adding instead of multiplying by the scale factor
Why it's wrong: Scale factor is a multiplicative relationship, not additive.
Correct: If scale factor is 2, multiply the original length by 2, don't add 2 to it.
Interactive Visual
Triangle Explorer
Area:
20000.0 sq units
Drag the vertices to change the triangle shape.
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Practice Problems
16 problemsTwo figures are similar if they have the same _____ but not necessarily the same _____.
Why It Matters
- Maps and blueprints: A map is similar to the actual land it represents
- Photography: Enlarging or reducing photos maintains similarity
- Architecture: Models of buildings are similar to the real structures
- Art: Artists use similarity to create perspective and depth
- Engineering: Scaled prototypes help test designs before full-scale production
Real World Applications
Map Reading
Maps use similarity to represent large areas on small paper. The scale tells you the ratio between map distance and actual distance.
Example:
If a map has a scale of 1:50,000, then 1 cm on the map represents 50,000 cm (or 500 m) in real life.
On a map with scale 1:25,000, two cities are 8 cm apart.
What is the actual distance between the cities in kilometers?
Step 1: Write the mathematical expression
Calculate: cm, then convert to km
Photo Enlargement
When you enlarge or shrink a photo, you create a similar figure. The aspect ratio stays the same.
Example:
A 4 by 6 inch photo enlarged by a factor of 2 becomes 8 by 12 inches. The ratio 4:6 = 8:12 = 2:3.
A photo is 10 cm wide and 15 cm tall. You want to enlarge it to 24 cm wide.
How tall will the enlarged photo be?
Step 1: Write the mathematical expression
Set up proportion:
Key Takeaways
- 1Similar figures have the same shape but not necessarily the same size
- 2Corresponding angles in similar figures are equal
- 3Corresponding sides in similar figures are proportional
- 4The scale factor is the ratio between corresponding sides
- 5Use proportions to find missing measurements in similar figures
Frequently Asked Questions
Glossary
- Similar figures
- Figures that have the same shape but not necessarily the same size
- Corresponding sides
- Sides in similar figures that are in the same relative position
- Corresponding angles
- Angles in similar figures that are in the same relative position
- Scale factor
- The ratio of corresponding sides between two similar figures
- Proportion
- An equation stating that two ratios are equal
Formula Card
Scale Factor
The ratio between corresponding sides of similar figures
Proportion
An equation showing that two ratios are equal
Cross-Multiplication
If two fractions are equal, their cross products are equal