Big picture
Geometrical construction makes shapes using tools and stated conditions. A rough sketch shows an idea, but a construction must meet requirements such as equal lengths, right angles or a fixed distance from a point. Careful tool use is as important as knowing the shape's name.
You can use construction to design a rectangular card, draw a circular pattern or locate a point equally far from two places. This chapter explains ruler, compass and set-square methods, with reasons that show why the results satisfy the requested conditions.
Learning goals
Use construction tools for their stated purposes.
Draw circles, rectangles and squares from given measurements.
Locate points satisfying two distance conditions.
Construct and explain perpendicular bisectors.
Verify shapes using lengths, angles and clear reasoning.
Core ideas
1. Tools, sketches and precise instructions
A ruler measures lengths and helps draw straight segments. A compass transfers a distance and draws circles or arcs. A set square supports right angles, and a protractor measures angles. Use the tools permitted by the question; a measured drawing and a strict ruler-and-compass construction are not always the same task.
A sketch need not be to scale. Label it first so you can see the required relationships. A construction then follows a repeatable sequence, retaining useful arcs and labels. Keep the compass point fixed while drawing a circle, and do not change its opening when transferring an equal length. State the tools and critical steps in words. A neat-looking result that was guessed does not show that the conditions were constructed.
2. Points at a fixed distance: circles
A circle is the set of points in a plane at a fixed distance from a centre. That distance is the radius. A diameter passes through the centre with endpoints on the circle and equals twice the radius.
To draw a circle of radius 3 cm, mark centre O, set the compass opening to 3 cm using the ruler, place the point at O and turn the pencil around once. Every point on the circumference is 3 cm from O. A diameter is therefore 2 × 3 = 6 cm. The centre is not on the circumference. A point inside is less than the radius from O; a point outside is farther away. These comparisons help explain intersection constructions rather than treating a circle as merely a decorative round outline.
3. Constructing and checking a rectangle
A rectangle has four right angles, with opposite sides equal and parallel. To construct rectangle ABCD with AB = 6 cm and BC = 4 cm, draw AB, construct perpendicular rays at A and B on the same side, mark AD = 4 cm and BC = 4 cm, then join C and D. Use an allowed right-angle tool or a justified perpendicular construction.
The remaining side CD is 6 cm because the construction makes opposite sides equal and parallel. Check the naming order around the boundary: A, B, C, D. A slanted quadrilateral with the correct side lengths is not necessarily a rectangle; the right angles matter. The perimeter is 6 + 4 + 6 + 4 = 20 cm. This calculation checks lengths but cannot by itself prove the angles are correct.
4. Squares and the equal-side condition
A square is a rectangle whose four sides are equal. To construct a square of side 4.5 cm, draw AB of that length, erect perpendiculars on the same side at A and B, mark AD and BC with the same compass opening, and join C to D. Check that the final side and every angle satisfy the definition.
A four-sided shape with all sides equal is not automatically a square: it can be a non-square rhombus. Conversely, four right angles alone allow unequal adjacent sides, giving a general rectangle. Both requirements matter. For side 4.5 cm, the perimeter is 4 × 4.5 = 18 cm. The diagonals of a square are equal and bisect each other, but use the requested side-and-angle construction rather than substituting an unproved visual guess.
5. Using compass arcs to locate a point
A compass arc represents possible positions at a specified distance from a centre. If C must be 4 cm from A and 3 cm from B, draw arcs of those radii centred at A and B. An intersection satisfies both distance conditions at once.
Suppose AB = 5 cm. Arcs with radii 4 cm and 3 cm can meet on either side of AB, giving two reflected choices for C. If the question requires C above AB, choose that intersection. Join A to C and B to C to display the required segments. A point on only one arc satisfies only one condition. Some choices of radii and separation give no intersection or just one, so do not claim that any two distance requirements always have two solutions. Explain what each arc guarantees.
6. Perpendicular bisectors and equal distances
A perpendicular bisector meets a segment at its midpoint and at a right angle. For segment AB, use equal compass radii greater than half AB to draw intersecting arcs above and below from A and B. Join the two intersections. The joining line is the perpendicular bisector.
The equal arcs make each intersection equally distant from A and B. Their joining line passes through the midpoint. If AB = 8 cm, that midpoint divides it into 4 cm and 4 cm. Choosing a radius too small prevents the required arc intersections. Preserve the same radius from both endpoints. A bisector alone means division into equal parts; perpendicular adds the right-angle requirement. When using this method to construct a rectangle, explain how the needed perpendicular is obtained rather than assuming every crossing line is at ninety degrees.
7. Circles, diagonals and distance comparisons
Diagonals join opposite vertices of a quadrilateral. In a rectangle, the diagonals are equal and meet at their common midpoint. In a square, they are also perpendicular. Do not transfer every square property to every rectangle.
A useful construction problem asks for a circle through the vertices of a rectangle. Its centre is the intersection of the diagonals, and the radius is the distance from that intersection to a vertex. Because the diagonals are equal and bisect each other, all four vertices are equally distant from the intersection. This explains the circle, rather than relying on an apparently successful compass turn. Check that the problem asks for a circle through the corners, not one touching the sides: these are different requirements and generally different constructions.
8. Construction problems, explanations and verification
A good solution contains a labelled sketch, ordered steps and a short reason. For a 6 cm by 4 cm rectangle, first specify the base, then the perpendicular directions, then the transferred lengths, and finally the closing side. Check that both perpendicular rays were drawn on the same side of the base; otherwise the intended boundary may be crossed.
Worked verification: AB and CD are each 6 cm; BC and AD are each 4 cm; all four corners are right angles. Therefore the named shape is the required rectangle. Its perimeter is 2 × (6 + 4) = 20 cm, which agrees with adding all four sides. Measurement is a useful final check, but a construction's reason explains why it should work even before small drawing errors are considered. Keep arcs visible if the task uses them as evidence, and do not erase labels needed to follow the sequence.
A reliable approach to questions
Read the command word before choosing an answer. A definition needs the distinguishing feature; a comparison needs matching criteria on both sides; an application needs a reason linked to the situation.
Extract the supplied evidence and assumptions. Keep a small working table rather than relying on memory. If information is absent, state the limit instead of inventing a value or claiming certainty.
Work through the relevant rule one step at a time. For a program, record the changing state; for a scenario, connect the requirement to the feature that meets it. Explain why the closest alternative fails.
Check the result against the original question, including boundary cases, units and wording. Present a concise final answer followed by enough reasoning for another student to reproduce your conclusion.
Connections that matter
Lines and angles explain perpendicularity and right angles.
Perimeter and area use the measurements of constructed shapes.
Symmetry explains reflected intersections and bisectors.
Common mistakes
Calling any equal-sided quadrilateral a square: right angles are also needed.
Changing compass width during equal-length transfer: preserve the radius.
Treating a sketch as proof: justify the construction conditions.
Assuming all rectangle diagonals are perpendicular: that is a square property, not a general rectangle property.
Revision checklist
I can use construction tools for their stated purposes.
I can draw circles, rectangles and squares from given measurements.
I can locate points satisfying two distance conditions.
I can construct and explain perpendicular bisectors.
I can verify shapes using lengths, angles and clear reasoning.