# Rectangles, rhombuses and squares · Years 8–9

> Explanation, worked examples and free practice for the national curriculum in England.

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

- Example: (A rectangle with its two diagonals dashed, meeting at a point.) In the rectangle ABCD the diagonals meet at O . The diagonal AC is 40 cm. How long is BO ?
- Answer: 20 cm
- Solution: BD = AC = 40 BO = 40 ÷ 2 = 20

### Diagonals of a rhombus

- Example: (A rhombus with its two diagonals dashed, meeting at a point.) 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: 10 cm
- Solution: AO = 12 ÷ 2 = 6 BO = 16 ÷ 2 = 8 AB² = 6² + 8² = 36 + 64 = 100 AB = 10

### Properties of a square

- Example: (A square with its two diagonals dashed, meeting at a point.) The diagonal of a square is 20 cm. What is the area of the square?
- Answer: 200 cm²
- Solution: 20 × 20 ÷ 2 = 200

### Which quadrilateral is it?

- Example: In the quadrilateral ABCD the diagonals are equal and bisect each other. Which quadrilateral is it for certain? Choose the most precise name.
- Options: Square · Rhombus · Rectangle · Parallelogram
- Answer: Rectangle
- Solution: 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. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).

[Practise on the site](https://kixplo.com/uk/en/) · [All Year 8 topics](https://kixplo.com/uk/en/maths/year-8/)