Definition and Properties of Similar Figures and Triangles
Understanding similarity is essential for excelling in school geometry, competitive exams like JEE and NTSE, and solving real-world measurement problems. It helps you recognize when two shapes have the same form but different sizes—a skill used in map-making, construction, and design. Building this concept will clarify many topics in maths and science.
Formula Used in Similarity
The standard formula is: \( \frac{\text{Side 1 of Figure A}}{\text{Corresponding Side 1 of Figure B}} = \frac{\text{Side 2 of Figure A}}{\text{Corresponding Side 2 of Figure B}} = \frac{\text{Side 3 of Figure A}}{\text{Corresponding Side 3 of Figure B}} \). This shows sides are proportional in similar figures.
Here’s a helpful table to understand similarity more clearly:
Similarity Table
| Figure | Corresponding Angles Equal? | Sides Proportional? | Are They Similar? |
|---|---|---|---|
| Triangle A & Triangle B | Yes | Yes | Yes |
| Rectangle & Square | Yes | No | No |
| Circle 1 & Circle 2 (any radius) | Yes | Yes | Yes |
This table shows how the pattern of similarity appears in geometric figures—look for equal angles and sides in the same ratio.
Worked Example – Solving a Similarity Problem
1. Two triangles have sides in proportion: the sides of Triangle PQR are 6 cm, 8 cm, and 10 cm, and the sides of Triangle XYZ are 9 cm, 12 cm, and 15 cm.2. Compare the ratios of corresponding sides:
3. Since all ratios are equal and the triangles have the same shape, they are similar by the SSS criterion.
4. If the smallest angle in Triangle PQR is 36°, then the smallest angle in Triangle XYZ is also 36° because corresponding angles of similar triangles are equal.
5. Therefore, these two triangles are similar with a scale factor of \( \frac{2}{3} \).
Practice Problems
- Check if triangles with sides 5 cm, 7 cm, 9 cm and 10 cm, 14 cm, 18 cm are similar.
- Are two circles of radius 3 cm and 7 cm similar?
- Find the scale factor between two squares with sides 4 cm and 10 cm.
- List two real-life objects that are similar figures.
Common Mistakes to Avoid
- Confusing similarity with congruence (which means same size and same shape).
- Forgetting to compare all corresponding sides and angles before concluding figures are similar.
- Assuming same shape always means same size.
Real-World Applications
The concept of similarity is useful in architecture, photography, map scaling, and model making. Engineers use it when designing scaled-down prototypes. Vedantu helps students connect similarity to practical uses in science and engineering.
We explored the idea of similarity, key properties, formulas, and solved step-by-step examples. Recognizing similar figures helps not only in geometry but also in many real-world scenarios. Practice more with Vedantu to master these concepts confidently.
Related reading: Deepen your understanding by exploring Similar Triangles.
FAQs on Similarity in Maths Concept Definition and Applications
1. What is similarity in mathematics?
Similarity in mathematics means that two figures have the same shape but not necessarily the same size. In geometry, similar figures have:
- Equal corresponding angles
- Proportional corresponding sides
2. What does it mean for two triangles to be similar?
Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional. This means:
- If triangle ABC ∼ triangle DEF, then ∠A = ∠D, ∠B = ∠E, ∠C = ∠F
- AB/DE = BC/EF = AC/DF
3. What are the criteria for similarity of triangles?
The three main criteria for similarity of triangles are AA, SAS, and SSS. These are:
- AA (Angle–Angle): Two corresponding angles are equal.
- SAS (Side–Angle–Side): Two pairs of sides are proportional and the included angle is equal.
- SSS (Side–Side–Side): All three pairs of sides are proportional.
4. How do you find the scale factor in similar figures?
The scale factor is the ratio of corresponding sides of two similar figures. To find it:
- Identify a pair of corresponding sides.
- Divide the length of one side by the matching side.
5. What is the difference between similar and congruent figures?
Similar figures have the same shape but different sizes, while congruent figures have the same shape and the same size. In summary:
- Similarity: Angles equal, sides proportional.
- Congruence: Angles equal, sides exactly equal.
6. How do you solve problems using similar triangles?
You solve similar triangle problems by setting up a proportion of corresponding sides. Steps:
- Verify triangles are similar using AA, SAS, or SSS.
- Write ratios of corresponding sides.
- Solve the resulting equation.
7. What is the symbol used for similarity?
The symbol used for similarity is ∼. For example, if triangle ABC is similar to triangle DEF, it is written as △ABC ∼ △DEF. The order of letters shows the correspondence of vertices.
8. How does area change in similar figures?
In similar figures, the ratio of areas is equal to the square of the scale factor. If the scale factor is k, then:
- Ratio of sides = k
- Ratio of areas = k²
9. Can you give an example of similar triangles in real life?
An example of similar triangles in real life is the use of shadows to measure height. When sunlight creates similar triangles:
- Height of object / Length of its shadow
- Height of person / Length of person's shadow
10. Why are corresponding angles equal in similar figures?
Corresponding angles are equal in similar figures because similarity preserves shape and angle measures. Even if the figure is enlarged or reduced (scaled), the angle sizes remain the same while side lengths change proportionally. This is why angle equality is a key condition for similarity.
