Rectangles, rhombuses and squares · Years 8–9
In Rectangles, rhombuses and squares (Years 8–9) you practise 4 skills: Diagonals of a rectangle · Diagonals of a rhombus · Properties of a square · Which quadrilateral is it?. A rectangle is a parallelogram with a right angle, so all its angles are right angles. Its diagonals are equal and bisect each other. In a right triangle the median to the hypotenuse is half the hypotenuse: it is half a diagonal of the rectangle that the triangle is half of.
A rhombus is a parallelogram with all sides equal. Its diagonals are perpendicular, bisect each other and bisect the angles. The area of a rhombus is half the product of its diagonals. A square is both a rectangle and a rhombus, so it has every property of both. To decide which quadrilateral you have, choose the most precise name the givens guarantee.
Tip: Every property of a parallelogram also holds in a rectangle, a rhombus and a square.
What you learn
Diagonals of a rectangle
For example: In the rectangle ABCD the diagonals meet at O . The diagonal AC is 40 cm. How long is BO ?
Answer and solution
The answer: 20 cm
BD = AC = 40 BO = 40 ÷ 2 = 20
Diagonals of a rhombus
For example: In the rhombus ABCD the diagonals meet at O . Given AC = 12, BD = 16 (in cm). How long is a side of the rhombus?
Answer and solution
The answer: 10 cm
AO = 12 ÷ 2 = 6 BO = 16 ÷ 2 = 8 AB² = 6² + 8² = 36 + 64 = 100 AB = 10
Properties of a square
For example: The diagonal of a square is 20 cm. What is the area of the square?
Answer and solution
The answer: 200 cm²
20 × 20 ÷ 2 = 200
Which quadrilateral is it?
For example: In the quadrilateral ABCD the diagonals are equal and bisect each other. Which quadrilateral is it for certain? Choose the most precise name.
- Square
- Rhombus
- Rectangle
- Parallelogram
Answer and solution
The answer: Rectangle
A quadrilateral whose diagonals bisect each other is a parallelogram. A parallelogram with equal diagonals is a rectangle. The diagonals are not given as perpendicular, so you cannot say it is a square. The answer: rectangle.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).