111
Point symmetry
Textbook: Part 4, lesson 111, pp. 10–12
GoalPoint (central) symmetry: a point symmetric about a centre; shapes with central symmetry.
New words
central (point) symmetry · markaziy simmetriyacentre of symmetry · simmetriya markazirotation · burishmidpoint · o‘rtasi
Explanation
To find the point A1 symmetric to A about a centre O, draw the ray AO and mark on the far side of O a segment equal to AO. Then O is the midpoint of AA1. If for every point of a shape the symmetric point also lies in the shape, the shape has central symmetry. Shapes with central symmetry: a segment, circle, square, rectangle, parallelogram; the letters N, S, Z, H, O, X. A triangle has no central symmetry.
Worked examples
A(1; 2), centre O(4; 5): O is 3 right and 3 up from A, so A1 is another 3 right and 3 up from O: (7; 8).
The point where the diagonals of a parallelogram cross is its centre of symmetry.
Class activity
“Half turn”: a paper shape is turned 180° (a half turn) about its centre with a pencil tip. If it lands exactly on itself, it has central symmetry.
Practice
1
AO = 5 cm and A1 is symmetric to A about the centre O. How many centimetres is AA1?
10
2
Write the coordinates of the point symmetric to A(1; 2) about the centre O(4; 5).
(7; 8)
3
Which of a circle, an equilateral triangle and a rectangle have central symmetry?
the circle and the rectangle
4
A parallelogram has a diagonal of 12 cm. The centre of symmetry splits the diagonal in two. How many cm is each piece?
6
5
Which of the letters A, N, S, T, X, Z have central symmetry?
N, S, X, Z