18
Centrally symmetric figures
Textbook: pp. 64–65
GoalRecognize centrally symmetric figures and find their centre.
New words
centrally symmetric figure · markaziy simmetrik shaklcentre of symmetry · simmetriya markaziparallelogram · parallelogrammcentre of a circle · aylana markazi
Explanation
If a figure lands on itself again when reflected in a point O, it is called centrally symmetric, and O is its centre of symmetry. A circle is centrally symmetric about its centre, because the two ends of every diameter are symmetric about the centre. A triangle has no such centre. Theorem: the centre of symmetry of a parallelogram is the point where its diagonals meet; so a rectangle, a rhombus and a square are centrally symmetric too. A figure made of two parallel lines has infinitely many centres: all of them lie on the parallel line running midway between the two lines.
Worked examples
The diagonals of parallelogram ABCD meet at O. About O, A goes to C and B goes to D, so the parallelogram maps onto itself.
The printed capitals H, N, S, Z, X are centrally symmetric (the centre is in the middle of the letter); F and T are not.
Class activity
“Find the centre”: turn a shape cut from paper (with an adult’s help) by 180° about a point and check whether it fits onto itself.
Practice
1
Does an equilateral triangle have a centre of symmetry?
No, a triangle is not centrally symmetric.
2
The diagonal AC of a parallelogram is 20 cm and the diagonals meet at O. The point symmetric to A about O is C. How many cm is AO?
10
3
How many centres of symmetry does a segment have?
1
4
Why is the intersection of the diagonals the centre of symmetry of a parallelogram?
The diagonals bisect each other, so the vertices are symmetric in pairs and the sides go onto the opposite sides.