Playing with Constructions · Lesson 3 of 8
Constructing Squares and Rectangles
“Build accurate squares and rectangles, and explain the purpose of every construction step.”
• Plan a construction from given side lengths. • Draw perpendiculars using a protractor and straight ruler. • Mark a length with a ruler or transfer it with a compass. • Construct and verify squares and rectangles in boundary order. • Explain why four right angles force equal opposite sides.
A plan gives every stroke a job
When side lengths are given, drawing a quadrilateral that merely looks right is not enough. Each side must have its intended length, and the corners must be right angles. A good plan establishes one condition at a time. Begin with an easy side, use it to create a right-angle direction, and then mark the next required length.
Gather a ruler, a protractor, a pencil, and a compass. A protractor measures angles; its centre must sit at the vertex and its baseline must line up with the chosen side. The ruler draws the straight line through the vertex and the angle mark. The compass is useful for copying a length accurately without reading that length again.
Two lines that meet at a right angle, 90°.
Lines in the same plane that never meet, however far they are extended. They have the same direction and stay the same distance apart.
To draw a perpendicular to a base at an endpoint, place the protractor centre on that endpoint. Align its zero line with the base and make a small point at 90° on the chosen side of the base. Remove the protractor and draw a straight ray from the endpoint through that point. A ray begins at a point and continues in one direction. Your 90° point guides the direction; it need not be the final corner.
A point used to guide a perpendicular sets the line direction. The actual vertex must also be at the required distance from the base endpoint. Measure or transfer that distance along the perpendicular after drawing it.
Constructing a square of side 6 cm
Let the square be PQRS in boundary order. We will put P at the lower left, Q at the lower right, R at the upper right, and S at the upper left. This keeps the labels consistent as we draw. The given information is a side length of 6 cm; the remaining conditions come from the definition of a square.
Problem
Construct square PQRS with every side 6 cm.
- 1.Draw base PQ = 6 cm. Label P and Q immediately, leaving room on one side for the square.
- 2.Draw a perpendicular ray to PQ at P. Mark S on it so that PS = 6 cm. Use a ruler, or set the compass to PQ and draw a short arc from P across the ray.
- 3.Draw a perpendicular ray to PQ at Q on the same side of the base. Mark R so that QR = 6 cm, using the same ruler length or compass opening.
- 4.Join S to R. The boundary order is P → Q → R → S → P. Verify that SR = 6 cm and all four corners are 90°.
Why is the top side the same length as the base? The two side rays have identical right-angle directions, so they stay the same distance apart. Marking equal heights on them puts the top corners level with one another. The segment joining those corners runs in the same direction as the base and spans the same separation. Its length is therefore 6 cm, and its two new corners are right angles.
You can see this by placing the construction against a rectangular sheet of paper or on a square grid: moving the same distance up two vertical lines keeps the endpoints on one horizontal level. The construction does not rely only on measuring the finished top side. The relationships built into its steps explain the result; measurement checks that your drawing followed those steps accurately.
Problem
How do you mark PS equal to PQ using a compass?
- 1.Place the pointed leg on P and adjust the pencil leg to Q. The opening now records the length PQ.
- 2.Without changing the opening, keep the point at P and draw a small arc across the chosen perpendicular ray. Call that intersection S.
- 3.Both Q and S were reached from P with the same opening, so PS = PQ. The ruler number is not needed to justify their equality.
Constructing a rectangle from two sides
For a rectangle, the heights on the two perpendicular rays must still agree with each other, but they need not match the base. This is the only change to the square plan. Choose one given length for the base and the other for the perpendicular sides. Mark both upright sides with the same second length and then join their endpoints.
Problem
Construct rectangle ABCD with AB = 6 cm and BC = 4 cm.
- 1.Draw AB = 6 cm. At A and B, construct perpendicular rays on the same side of AB.
- 2.Mark D at distance 4 cm from A and C at distance 4 cm from B. Copy the 4 cm length with a compass if you prefer.
- 3.Join D and C. The final figure has AB = CD = 6 cm and AD = BC = 4 cm.
- 4.Check all four angles with a protractor. Read the boundary name A → B → C → D, rather than jumping across AC or BD.
Problem
Construct PQRS with PQ = 10 cm and QR = 2 cm.
- 1.Draw PQ = 10 cm and create perpendicular rays at P and Q. Leave enough room for the 2 cm height.
- 2.Mark S with PS = 2 cm and R with QR = 2 cm on the same side of PQ. Use one compass opening for both heights.
- 3.Join S to R. The opposite pairs should be PQ = SR = 10 cm and PS = QR = 2 cm, with all four angles 90°.
- 4.The figure is a rectangle even though it is narrow. Its adjacent side lengths differ, so it is not a square.
The longer given length need not be horizontal. You could begin with the 4 cm side of the first rectangle, or place a base at a slant. The same perpendicular and equal-length steps would produce the same side relationships. A rotated construction should be checked using a protractor, rather than judged against the edge of the page.
Checking the drawing and its logic
A completed drawing needs two kinds of checks. First, check the conditions you deliberately set: the base length, the perpendicular directions, and the two equal copied heights. Then check the resulting conditions: the closing side and remaining angles. If something is wrong, trace the mistake back to a step rather than forcing the last line into place.
| Check | What to compare | Possible cause of an error |
|---|---|---|
| Base | Given length and measured length | Wrong ruler endpoints |
| Side directions | Both 90° to the base | Protractor centre or baseline misaligned |
| Heights | Equal required lengths on both rays | Compass opening changed |
| Closing side | Same length as the base | Unequal heights or non-perpendicular rays |
| Naming | Each consecutive pair shares a boundary side | Labels assigned inconsistently |
Small drawing errors are expected when measuring by hand. A side that reads just slightly differently does not overturn the mathematical construction. However, a visible tilt in the closing side or a large difference in the heights shows that a condition was not set correctly. Repeat that part carefully. State the intended exact lengths separately from the approximate measurements of your pencil drawing.
Problem
A student draws a 6 cm base and two perpendicular sides, but sets heights of 4 cm and 5 cm. What fails?
- 1.The perpendicular directions are correct, but the intended opposite sides have different lengths.
- 2.Their upper endpoints are at different heights, so joining them gives a slanting closing side. Its corners cannot both be the required right angles.
- 3.Remark the 5 cm height as 4 cm before drawing the final side. Do not try to repair the shape by changing only the closing line.
Can four right angles have unequal opposite sides?
Try to draw a simple quadrilateral with four right angles but unequal opposite sides. Starting with a base, its two neighbouring sides must be perpendicular to it. To make the top two corners right angles as well, the closing side must run in the same direction as the base. This requires the two side heights to match.
The fixed separation between the two upright sides gives equal top and bottom lengths. Their common height gives equal left and right lengths. Thus four right angles already force the rectangle side relationships. You cannot construct the proposed exception. Drawing a rough “almost rectangle” with one slanted side would change the angle condition, not solve the problem.
If one height is drawn above the base and the other below it, joining their endpoints does not make the intended rectangle. Decide the interior side first and put both height points there.
Quiz
After drawing PQ = 6 cm for a square, where must S lie?
Which instrument directly copies a segment length without needing its numerical value?
For a rectangle with base 8 cm and height 3 cm, what heights should be marked on the two side rays?
Why does marking equal heights on perpendicular side rays help?
Can a simple quadrilateral have four right angles but unequal opposite sides?
What is the first correction if the closing side slopes because the two side heights differ?
Practice Problems
- Construct a square of side 6 cm and write the purpose of each step.
- Construct a square of side 3.5 cm using a compass to transfer its side length. Check all sides and angles.
- Construct a 4 cm by 6 cm rectangle in boundary order and record both opposite side pairs.
- Construct a 2 cm by 10 cm rectangle. Turn the page and explain why the classification is unchanged.
- Draw an unmeasured segment and construct a square using that segment as a side. Explain every equality without a numerical length.
- Construct a 5 cm by 3 cm rectangle on a slanting base. Use a protractor to verify the corners.
- A student sets the two side heights as 3 cm and 4 cm. Sketch the result and explain why it cannot have four right angles.
- Explain why a quadrilateral with four right angles and unequal opposite sides cannot be constructed.
A 6 cm square needs four 6 cm sides and four right angles. A 4 cm by 6 cm rectangle has opposite sides 4 cm and 6 cm in matching pairs; a 2 cm by 10 cm rectangle works the same way.
Key Takeaways
• Begin with given information and establish one condition at each step. • Perpendicular directions come from 90° angles; distances locate the actual vertices on those directions. • A compass transfers a length without changing it. • Equal heights on perpendicular side rays complete a rectangle; matching the height to the base gives a square. • Check the final sides, angles, and boundary-order labels. • Four right angles cannot coexist with unequal opposite sides in a simple quadrilateral.