8
The square and its properties
Textbook: pp. 25–26
GoalKnow the definition and properties of a square and link it to the parallelogram, rectangle and rhombus.
New words
square · kvadratdiagonal · diagonalto bisect an angle · burchakni teng ikkiga bo‘lishequilateral · teng tomonli
Explanation
A rectangle with equal sides is called a square; in other words, a square is a rhombus with right angles. A square is at once a parallelogram, a rectangle and a rhombus, so it has all their properties. All angles of a square are right. Its diagonals are equal, perpendicular, bisect each other and bisect the angles of the square. So a diagonal makes a 45° angle with a side. A rhombus with equal diagonals is a square, because a parallelogram with equal diagonals is a rectangle.
Worked examples
The perimeter of a square is 36 cm. A side is 36 : 4 = 9 cm.
The side of a square is 14 cm. The distance from the intersection O of the diagonals to a side is half the side, 7 cm.
Class activity
“The shape family”: draw on the board nested boxes for parallelogram, rectangle, rhombus and square and write the properties of each.
Practice
1
The side of a square is 8 cm. What is its perimeter (cm)?
32
2
The perimeter of a square is 52 cm. How long is a side (cm)?
13
3
What angle does a diagonal of a square make with a side (degrees)?
45
4
Why is a rhombus with equal diagonals a square?
A rhombus is a parallelogram, a parallelogram with equal diagonals is a rectangle, and a rectangle with equal sides is a square.