108
Symmetrical shapes
Textbook: Part 3, lesson 108, pp. 120–122
GoalSymmetrical shapes: several axes, reflecting a shape in a line, symmetrical numbers.
New words
reflection · akslantirishcommon region · umumiy sohapalindromic number · simmetrik sonto intersect · kesishmoq
Explanation
Some shapes have several axes of symmetry: a square has 4, a rectangle 2, a disc infinitely many. To reflect a shape in an axis we draw a perpendicular from each vertex to the axis and measure the same distance on the other side. Two circles whose centres are closer than the sum of their radii intersect; their common region is a symmetrical shape with two axes. Numbers can be symmetrical too: 454 and 12 321 read the same left to right and right to left.
Worked examples
Circles centred at A and B have radius 3 cm and AB = 5 cm. Since 5 < 3 + 3 = 6 the circles intersect; the common region is symmetrical about the line AB and about the perpendicular through the midpoint of AB.
A rectangle with area 36 cm² can have sides 4 cm and 9 cm: perimeter 2 × (4 + 9) = 26 cm. With sides 6 cm and 6 cm (a square): 2 × (6 + 6) = 24 cm.
Class activity
“Magic mirror”: in pairs, one child draws half a shape on squared paper and the other draws its symmetrical half. They fold the page to check that the halves coincide.
Practice
1
Two circles of radius 3 cm intersect and their common region is a symmetrical shape. How many axes of symmetry does it have?
2
2
A rectangle of area 36 cm² has sides 4 cm and 9 cm. What is its perimeter in centimetres?
26
3
A symmetrical shape has area 36 cm². The axis divides it into two equal parts. What is the area of one part in square centimetres?
18
4
A vertical axis of symmetry passes through 5 on the OX axis. Write the point symmetrical to (3; 2) about this axis.
(7; 2)
5
232 and 454 are symmetrical numbers. What is their sum?
686