6
Similarity of polygons
Textbook: pp. 28–29
GoalKnow the definition of similar polygons and the similarity coefficient, and find unknown lengths from proportional corresponding sides.
New words
similar polygons · o‘xshash ko‘pburchaklarcorresponding sides · mos tomonlarproportional · proporsionalsimilarity coefficient · o‘xshashlik koeffitsiyenti
Explanation
Shapes that look the same but differ in size, for example a small and a large print of one photograph, are called similar. Two polygons are similar polygons if all corresponding angles are equal and all corresponding sides are proportional; similarity is written with the sign ∽. The ratio k = A₁B₁ : AB = B₁C₁ : BC = … is the similarity coefficient. Equal angles alone are not enough: all angles of a rectangle are right, but 2 × 4 and 3 × 4 rectangles are not similar. Proportional sides alone are not enough either: a square and a rhombus can have proportional sides but different angles.
Worked examples
Rectangles 4 × 6 and 6 × 9: 6 : 4 = 9 : 6 = 3/2 and all angles are right, so they are similar with k = 3/2.
A quadrilateral has sides 3, 5, 6, 8 cm. A similar one with k = 2 has sides 6, 10, 12, 16 cm.
Class activity
“Enlarging”: on squared paper draw a small triangle and then a similar one with k = 3; compare corresponding angles with a protractor.
Practice
1
A triangle has sides 4, 6 and 8 cm. A similar triangle has shortest side 6 cm. How long is its longest side (cm)?
12
2
ABCD ∽ A₁B₁C₁D₁, AB = 10 cm and A₁B₁ = 25 cm. Find the similarity coefficient.
5/2 (that is, 2.5)
3
Are the rectangles 2 × 5 and 4 × 11 similar?
No, because 4 : 2 = 2 but 11 : 5 ≠ 2.
4
Why are a square and a non-square rectangle never similar?
Their angles are equal, but a square has side ratio 1 and the rectangle does not, so the sides are not proportional.