Similarity of Triangles criteria formula and solved examples
The concept of Similarity of Triangles is a crucial topic in Maths, allowing you to compare the shape, side lengths, and angle measurements of two triangles. It has practical uses in geometry, map-reading, architecture, and problem solving for exams such as CBSE Class 9, 10, JEE, and more.
What Is Similarity of Triangles?
Similarity of triangles means two triangles have exactly the same shape, but not necessarily the same size. In detail, triangles are similar if their corresponding angles are equal, and their corresponding sides are in the same proportion. This concept is widely used in identifying scale drawings, solving geometry problems, and understanding the relationship between similar and congruent figures.
Key Formula for Similarity of Triangles
The key formula for similarity of triangles is:
If △ABC ∼ △DEF, then
\[
\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}
\]
and
\[
\angle A = \angle D, \quad \angle B = \angle E, \quad \angle C = \angle F
\]
Criteria for Similarity of Triangles (AA, SSS, SAS)
To quickly test if two triangles are similar, check these three main rules:
| Criterion | What to Check | How to Apply |
|---|---|---|
| AA (Angle-Angle) | Two pairs of corresponding angles are equal. | If ∠A=∠D and ∠B=∠E, then the triangles are similar. |
| SSS (Side-Side-Side) | All three pairs of corresponding sides are in the same ratio. | If \( \frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} \), then the triangles are similar. |
| SAS (Side-Angle-Side) | Two pairs of sides are in the same ratio and the included angle is equal. | If \( \frac{AB}{DE} = \frac{AC}{DF} \) and ∠A=∠D, then the triangles are similar. |
Quick Trick: Remember "AA, SSS, SAS" — focus on angles and proportional sides!
Proving Similarity – Step-by-Step Example
Example: Show that triangles ABC and DEF (where AB=6 cm, BC=8 cm, CA=10 cm, DE=9 cm, EF=12 cm, FD=15 cm) are similar.
\( \frac{AB}{DE} = \frac{6}{9} = \frac{2}{3} \)
\( \frac{BC}{EF} = \frac{8}{12} = \frac{2}{3} \)
\( \frac{CA}{FD} = \frac{10}{15} = \frac{2}{3} \)
2. Since all three side ratios are equal, by the SSS criterion, triangles ABC and DEF are similar.
Properties of Similar Triangles
- Corresponding angles are equal.
- Corresponding sides are in the same ratio (proportional).
- The area ratio of two similar triangles equals the square of the scale factor for corresponding sides.
- If one triangle is congruent to another, it is also similar.
Difference Between Similar and Congruent Triangles
| Similar Triangles | Congruent Triangles |
|---|---|
| Same shape, can have different sizes. | Same shape and same size. |
| All corresponding angles equal; sides proportional. | All corresponding angles and sides equal. |
| Symbol: ∼ (e.g., △ABC ∼ △DEF) | Symbol: ≅ (e.g., △ABC ≅ △DEF) |
Classroom Tip
A handy mnemonic: “AA, SSS, SAS” helps you remember the similarity of triangles rules. Pair it with a simple diagram in class or revision notes! Vedantu teachers often draw colored triangles side-by-side to help you spot similarities faster.
Try These Yourself
- Are triangles with angles 65°, 55°, 60° and 65°, 60°, 55° similar?
- The sides of a triangle are 5 cm, 12 cm, 13 cm. Another triangle has sides 10 cm, 24 cm, 26 cm. Are they similar?
- Find the value of x if two similar triangles have corresponding sides of length 4 cm and 6 cm, 6 cm and x cm.
Frequent Errors and Misunderstandings
- Forgetting to check all angle pairs or side ratios.
- Mixing up similarity and congruence (not every similar triangle is congruent).
- Not matching the correct order of corresponding vertices.
Real-Life Applications
- Creating maps and scale models in geography.
- Designing ramps, roofs, and art using geometric patterns in architecture.
- Measuring the height of big objects using shadows (indirect measurement).
- Solving image enlargement/shrinking problems in computer graphics.
Relation to Other Concepts
The topic of Similarity of Triangles links directly to Triangle Theorems, and broader Polygons and Their Properties. Mastering it helps in both coordinate and practical geometry chapters.
We explored Similarity of Triangles, its rules, formulas, solved examples, and connections to real-world situations. Keep practicing with Vedantu to become a triangle similarity pro!
FAQs on Similarity of Triangles Complete Guide with Theorems and Applications
1. What is similarity of triangles in geometry?
The similarity of triangles means that two triangles have the same shape but not necessarily the same size. In similar triangles:
- Corresponding angles are equal.
- Corresponding sides are proportional.
2. What are the conditions for similarity of triangles?
The three main conditions for similarity of triangles are AA, SAS, and SSS similarity criteria. These are:
- AA (Angle-Angle): Two corresponding angles are equal.
- SAS (Side-Angle-Side): Two sides are proportional and the included angle is equal.
- SSS (Side-Side-Side): All three pairs of corresponding sides are proportional.
3. How do you prove two triangles are similar?
To prove two triangles are similar, show that they satisfy AA, SAS, or SSS similarity criteria. Follow these steps:
- Compare corresponding angles to check if at least two are equal (AA).
- Or verify that two sides are proportional and the included angle is equal (SAS).
- Or check if all three corresponding sides are proportional (SSS).
4. What is the formula for similar triangles?
The key formula for similar triangles is that the ratio of corresponding sides is equal, called the scale factor. If △ABC ~ △DEF, then:
- AB/DE = BC/EF = AC/DF
5. How do you find a missing side in similar triangles?
To find a missing side in similar triangles, use the property that corresponding sides are in the same ratio. Steps:
- Write the proportion of corresponding sides.
- Substitute known values.
- Solve for the unknown.
6. What is the difference between similar and congruent triangles?
The difference is that similar triangles have the same shape but different sizes, while congruent triangles have the same shape and the same size. In similar triangles:
- Angles are equal.
- Sides are proportional.
- Angles are equal.
- Corresponding sides are exactly equal in length.
7. Why is AA enough to prove triangle similarity?
AA is enough because if two angles of one triangle are equal to two angles of another, the third angles must also be equal. Since the sum of angles in a triangle is 180°, the third angle automatically matches. With all corresponding angles equal, the triangles have the same shape, which proves similarity.
8. What is the scale factor in similar triangles?
The scale factor in similar triangles is the constant ratio of corresponding sides. It is calculated as:
- Scale Factor = (Side of larger triangle) ÷ (Corresponding side of smaller triangle)
9. Can you give an example of similar triangles?
An example of similar triangles is when two triangles have sides in the ratio 2:4:6 and 4:8:12. Since:
- 2/4 = 4/8 = 6/12 = 1/2
10. What are the properties of similar triangles?
The main properties of similar triangles are equal angles and proportional sides. Key properties include:
- Corresponding angles are equal.
- Corresponding sides are proportional.
- The ratio of their areas equals the square of the scale factor.
- The ratio of their perimeters equals the scale factor.
