17
Central symmetry and rotation
Textbook: pp. 52–57
GoalKnow central symmetry and rotation, find coordinates of a point symmetric about a centre, and recognise centrally symmetric figures.
New words
central symmetry · markaziy simmetriyacentre of symmetry · simmetriya markazirotation · burishangle of rotation · burish burchagi
Explanation
Points A and A₁ are symmetric about O if O is the midpoint of AA₁. Sending every point to its symmetric point about O is central symmetry; it is a motion. About the origin A(x; y) goes to A₁(−x; −y), and in general for a centre O(a; b) to A₁(2a − x; 2b − y). A figure that maps to itself about O is centrally symmetric: the parallelogram, rectangle, rhombus, square and circle are; a triangle and a trapezoid are not. In a rotation about O through φ we have OA = OA₁ and ∠AOA₁ = φ; a rotation is a motion, and a rotation through 180° is central symmetry.
Worked examples
About the centre O(3; 1) the point A(5; 4) goes to x₁ = 2·3 − 5 = 1, y₁ = 2·1 − 4 = −2, that is A₁(1; −2).
A parallelogram is centrally symmetric about the intersection O of its diagonals: A goes to C and B goes to D, since O is the midpoint of the diagonals.
Class activity
“Paper spinner”: rotate a letter 180° about its centre and find which letters stay unchanged (for example H, N, O, S, X, Z).
Practice
1
What is the x-coordinate of the point symmetric to M(−6; 2) about the origin?
6
2
Find the point symmetric to A(4; 5) about the centre O(1; 2).
(−2; −1)
3
Which is not centrally symmetric: square, trapezoid, parallelogram, circle?
The trapezoid
4
Why is a rotation through 180° the same as central symmetry?
At 180° the points A, O and A₁ are collinear with OA = OA₁, so O is the midpoint of AA₁.