☰ Contents · Geometry

Homothety and constructing similar polygons

Lessons 20–21 · 2 lessons · B. Xaydarov, E. Sariqov, A. Qo‘chqorov. Geometry, Grade 9 (textbook for general secondary schools), revised 4th edition. Huquq va Jamiyat Publishing, Tashkent, 2019
20

Homothety and similarity

Textbook: pp. 62–63
GoalKnow the definition of homothety (centre, coefficient), understand that it is a similarity transformation, and use it in coordinates.
New words
homothety · gomotetiyacentre of homothety · gomotetiya markazicoefficient of homothety · gomotetiya koeffitsiyentihomothetic figures · gomotetik shakllar
Explanation

Let a point O and a number k > 0 be given. If for each point X of figure F we take on ray OX the point X₁ with OX₁ = k·OX, the resulting transformation is the homothety with centre O and coefficient k. Theorem: a homothety is a similarity transformation, that is X₁Y₁ = k·XY. Proof: in triangles OXY and OX₁Y₁ the angle O is common and OX₁ : OX = OY₁ : OY = k, so they are similar by SAS. If 0 < k < 1 the figure shrinks, if k > 1 it stretches, and with k = 1 it stays the same. With the centre at the origin, the point A(x; y) goes to (kx; ky). A figure homothetic to a circle is a circle, and circles tangent to the sides of an angle are related by a homothety centred at the vertex of the angle.

Worked examples
Centre at the origin, k = 3: A(2; −1) → A₁(6; −3). OA = √5 and OA₁ = 3√5.
Centre at the origin, k = 1/2: B(8; 6) → B₁(4; 3). OB = 10 and OB₁ = 5, so OB₁ = OB/2.
Class activity

“Projector”: with an adult's supervision, cast the shadow of a cardboard shape on a wall with a lamp or phone torch, move it away from the light and watch how the shadow's size changes.

Practice
1
The centre is the origin and k = 4. Where does C(3; −2) go? Give the x-coordinate.
2
In a homothety OX = 7 cm and k = 3. How long is OX₁ (cm)?
3
In a homothety a triangle of area 5 cm² is mapped with k = 6. What is the new area (cm²)?
4
Why does a homothety preserve angles?