Rectangle and square properties and construction

Use properties to construct the shape

A rectangle has four right angles and equal opposite sides. A square has these properties and four equal sides. A square is therefore a special rectangle, even when it is turned.

The opposite sides of a rectangle are also parallel: they stay the same distance apart and do not meet, even when extended. Neighbouring sides meet at right angles, so they are perpendicular. Equal length and parallel direction are different properties. A short side can be parallel to another short side without being parallel to either long side.

In rectangle ABCD, AB and CD are opposite sides, as are BC and AD. Therefore AB = CD and BC = AD. An equals sign here compares lengths. The parallel pairs are AB ∥ CD and BC ∥ AD. The four neighbouring pairs are perpendicular: AB ⊥ BC, BC ⊥ CD, CD ⊥ AD and AD ⊥ AB. Each pair meets at one of the four right-angled corners.

Naming the side in reverse does not change it: AD and DA name the same side. A line from A to C would go across the inside; it is not a side of the boundary.

Check your understanding 1 Which properties change for a square?

For square EFGH, name every parallel pair and every perpendicular pair of sides. Which sides have equal lengths? Explain why equal length does not make two neighbouring sides parallel.

Your side pairs and explanation

Show answer

The parallel pairs are EF ∥ GH and FG ∥ EH. The perpendicular pairs are EF ⊥ FG, FG ⊥ GH, GH ⊥ EH and EH ⊥ EF. All four sides have equal lengths: EF = FG = GH = EH. This equality chain means any pair chosen from those four sides has equal lengths.

EF and FG are equal but perpendicular, because they meet at a right angle. Parallel describes direction, not length. Turning the square preserves all these relationships.

When listing pairs in a check, you may type || for ∥ and perp for ⊥. Separate pairs with semicolons. For equal lengths, either list every equal pair or use a chain joining all the equal sides.

To draw an 8 cm by 5 cm rectangle, draw an 8 cm base. At each end, use a set square to draw a perpendicular on the same side of the base. Mark 5 cm along each perpendicular and join the marks. Check the fourth side and the right angles. A drawing that merely looks rectangular is not a measurement check.

For a square of side 6 cm, use the same method with both dimensions 6 cm. These constructions use side and angle properties; no diagonal property is needed.

Build and check a closed boundary

A rectangle is more than two suitable side lengths. The sides must join to enclose a region, and the corners must be right angles. Plan the four corners in order around the boundary before joining them.

Try the 8 cm by 5 cm rectangle below. On this canvas, 10 drawing units represent 1 cm. A side of 8 cm therefore spans 80 units, and a side of 5 cm spans 50 units. The canvas represents lengths; its size on the screen can change.

Choose Solid line, tap each of your four corners in order, then use Finish as loop to join the last corner back to the first. To work by keyboard, open Type coordinates instead, choose Joined points, enter each x, y pair on its own line, and choose Closed path before adding it. Enter only your corners; the closing side is joined for you.

Try the model. You can change your answer or look at the worked model. This activity has no score.

The check looks for one closed boundary, four right angles and the exact requested side lengths, using 10 drawing units for 1 cm. It checks the finished geometry; practise ruler and set-square placement on paper too. You can also draw on paper and compare with the public answer.

Show solution

Coordinates increase to the right and down the page. Lengths use equal coordinate scales.

Points and lines in the drawing

Object	Values	Given or drawn
Boundary	Closed boundary: (20, 30) → (100, 30) → (100, 80) → (20, 80) → (20, 30)	Drawn

If you placed the sides along the grid, check the horizontal and vertical gaps between adjacent corners. An 8 cm by 5 cm outline must use both requested lengths, in alternating order. Check closure and every right angle as well as the lengths.

Check the properties, not the orientation

A learner draws the four-sided figure above with opposite sides equal in pairs: one pair is 8 cm and the other is 5 cm. There are no right angles. Is this enough to make it a rectangle?

Your reason

A square has the rectangle properties and equal sides

Draw a square of side 5 cm below. Each side must span 50 drawing units. You can place its sides along the grid. Choose its position yourself and check all four corners as well as the side lengths.

Try the model. You can change your answer or look at the worked model. This activity has no score.

The check looks for one closed boundary, four right angles and the exact requested side lengths, using 10 drawing units for 1 cm. It checks the finished geometry; practise ruler and set-square placement on paper too. You can also draw on paper and compare with the public answer.

Show solution

Coordinates increase to the right and down the page. Lengths use equal coordinate scales.

Points and lines in the drawing

Object	Values	Given or drawn
Boundary	Closed boundary: (70, 30) → (100, 70) → (60, 100) → (30, 60) → (70, 30)	Drawn

The worked model is turned. To check a sloping side on paper, line up a ruler with that side, then use a set square at the corner. Its horizontal and vertical changes do not separately give its length. Turning a square changes those changes while keeping the side lengths and right angles.

Try a new size on paper

Construct a square of side 4 cm. Check all four sides and corners. Turn the page and explain why it is still a square.