≡Mathematics EN 9th grade: Similar Figures
1. What are similar figures
Two figures are similar when:
✔ They have the same shape
✔ Corresponding angles are equal
✔ Corresponding sides are proportional
👉 There is a similarity ratio \( k \)
2. Similarity of triangles
Two triangles are similar if they satisfy one of the criteria:
✔ Criterion SSS (side-side-side)
All three sides are proportional:
\( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \)
✔ Criterion SAS (side-angle-side)
Two sides proportional and the included angle equal:
\( \frac{a_1}{a_2} = \frac{b_1}{b_2} \) and equal angle
✔ Criterion AA (angle-angle)
Two angles are equal:
✔ The third will also be equal automatically
3. Example of similarity
Triangles with sides:
\( 3, 4, 5 \) and \( 6, 8, 10 \)
👉 Checking:
\( \frac{6}{3} = 2, \quad \frac{8}{4} = 2, \quad \frac{10}{5} = 2 \)
✔ They are similar
✔ Similarity ratio: \( k = 2 \)
4. Ratio of perimeters
The perimeter changes in the same ratio \( k \):
\( \frac{P_2}{P_1} = k \)
Example:
If \( k = 3 \):
✔ The perimeter triples
5. Ratio of areas
The area changes with the square of the ratio:
\( \frac{A_2}{A_1} = k^2 \)
Example:
If \( k = 2 \):
✔ \( A_2 = 4A_1 \)
6. Complete example
Two similar triangles with \( k = 3 \)
✔ If the smaller perimeter is \( 10 \):
\( P_2 = 3 \cdot 10 = 30 \)
✔ If the smaller area is \( 5 \):
\( A_2 = 5 \cdot 3^2 = 45 \)
7. Important interpretation
✔ Sides → multiply by \( k \)
✔ Perimeter → multiply by \( k \)
✔ Area → multiply by \( k^2 \)
8. Key points for the exam
✔ Identify similar triangles
✔ Apply SSS, SAS, and AA criteria
✔ Find the ratio \( k \)
✔ Relate perimeters and areas correctly
Did you know?
