☰ Contents · Geometry

Central symmetry

Lessons 17–18 · 2 lessons · A. A. Rahimqoriyev, M. A. To‘xtaxo‘jayeva. Geometry, Grade 8 (textbook for general secondary schools), revised 3rd edition. Yangiyo‘l Poligraf Servis, Tashkent, 2014
17

Central symmetry and its properties

Textbook: pp. 61–63
GoalConstruct central symmetry about a point and know its properties.
New words
central symmetry · markaziy simmetriyacentre of symmetry · simmetriya markaziorigin · koordinatalar boshisymmetric point · simmetrik nuqta
Explanation

If a point O cuts the segment AA1 into two equal halves, the points A and A1 are called symmetric about O, and O is the centre of symmetry. The point O corresponds to itself. If reflecting every point of a figure F in O gives a figure F1, then F and F1 are called centrally symmetric about O. Theorem 1: central symmetry does not change distances between points, and angles keep their size. Theorem 2: a segment goes to a segment, a ray to a ray, a line to a line; a line not passing through the centre goes to a line parallel to it. About the origin the point (x; y) goes to (−x; −y).

Worked examples
The image of A(3; −5) under symmetry about the origin is A1(−3; 5).
Segment AB with A(1; 2), B(4; 3). About the origin the images are A1(−1; −2), B1(−4; −3). Segment A1B1 is equal and parallel to AB.
Class activity

“Upside-down drawing”: mark a centre on squared paper and draw the image of a triangle as if it were turned by 180° about that point.

Practice
1
Write the image of (−7; 2) under symmetry about the origin.
2
A point is 5 cm from the centre of symmetry. How far is it from its symmetric point (cm)?
3
What is the abscissa of the image of A(2; −9) under symmetry about the origin?
4
Why does a line through the centre map onto itself under central symmetry?