Primary & Middle School Mathematics · Grades 1–9

33Geometric Constructions

Ruler, set square, compass: with these three tools and a bit of method, any figure described in words can be built exactly. This chapter constructs perpendiculars, parallels, triangles and the special quadrilaterals — craft first, theory later (Chapters 40 and 41).

33.1 Perpendiculars and parallels through a point

Method 33.1 (The two basic constructions)

Perpendicular to $d$ through $A$ : slide the set square along $d$ until its other right-angle edge reaches $A$ ; draw; mark the right angle.

Parallel to $d$ through $A$ : press a ruler against the set square as a rail laid along $d$ ; slide the set square up to $A$ ; draw. (Check: the gap between the two lines is the same everywhere.)

Example 33.2 (Chaining constructions)

Many figures are chains of these two moves. A rectangle $ABCD$ with $AB = 5$ cm and $BC = 3$ cm: draw $[AB]$ ; perpendicular at $A$ ; perpendicular at $B$ ; mark $D$ and $C$ at $3$ cm on the same side; join. Every step is one basic construction.

33.2 Triangles with the compass

Method 33.3 (Triangle from three sides)

To build a triangle with sides $6$ cm, $5$ cm, $4$ cm:

draw the longest side, $[AB]$ with $AB = 6$ cm;
arc of center $A$ and radius $5$ cm;
arc of center $B$ and radius $4$ cm;
their crossing is the third vertex $C$ ; join.

The compass guarantees the lengths: every point of the first arc is at exactly $5$ cm from $A$ .

Two arcs, one crossing: the triangle 6–5–4 built with the compass. (If the two short sides together were shorter than the base, the arcs would not meet — more on this in .) — Two arcs, one crossing: the triangle $6$ – $5$ – $4$ built with the compass. (If the two short sides together were shorter than the base, the arcs would not meet — more on this in Chapter 51.)

Example 33.4 (Isosceles and equilateral for free)

Same compass opening for both arcs $\Rightarrow$ isosceles triangle. Opening equal to the base $\Rightarrow$ equilateral. The compass does not measure: it copies lengths, which is even better.

33.3 The special quadrilaterals, by construction

Method 33.5 (Building the family)

Rectangle: two pairs of parallels crossing at right angles — or the chain of Example 33.2;
square: a rectangle whose four sides use the same length — carry it with the compass;
rhombus: four equal sides — draw one side, then two compass arcs from its ends with the same opening fix the other two vertices (a rhombus is two isosceles triangles glued along a diagonal).

After building, always check with the tools: set square on the corners, compass on the sides.

Example 33.6 (Recognizing from a description)

“A quadrilateral with four $4$ cm sides and one right angle” — build it: the four equal sides force a rhombus shape, and pressing one corner to a right angle straightens all the others: the result is a square. Some descriptions secretly describe a more special shape than they announce. (Why the right angle spreads to all four corners is explained in Chapter 52.)

33.4 Circles in constructions

Example 33.7 (The circle as a tool)

“Find all points at $3$ cm from $A$ ”: the circle of center $A$ , radius $3$ cm. “Find all points at $3$ cm from $A$ and $4$ cm from $B$ ”: draw both circles; their crossings (two, one or zero points) are the answers. Treasure maps and triangle constructions use exactly this idea.

33.5 Exercises

Exercise 33.1 ★

Draw a line $d$ and a point $A$ at about $3$ cm from it. Construct the perpendicular to $d$ through $A$ , then the parallel to $d$ through $A$ . Mark the right angles.

Solution

Solution of Exercise 33.1.

Both constructions follow Method 33.1; the perpendicular and the parallel through $A$ are themselves perpendicular to each other.

Exercise 33.2 ★

Construct a rectangle $ABCD$ with $AB = 7$ cm and $BC = 4$ cm, following Example 33.2. Check the fourth angle with the set square: if the construction is exact, it is right automatically.

Solution

Solution of Exercise 33.2.

Chain of perpendiculars as in Example 33.2. If the three built angles are exactly right and the lengths exact, the fourth angle has no choice but to be right too — a well-built rectangle checks itself.

Exercise 33.3 ★

Construct a triangle with sides $7$ cm, $6$ cm and $3$ cm (Method 33.3).

Solution

Solution of Exercise 33.3.

Base $[AB]$ of $7$ cm; arcs of radii $6$ cm (from $A$ ) and $3$ cm (from $B$ ); the crossing is $C$ .

Exercise 33.4 ★

Construct an isosceles triangle with base $5$ cm and equal sides of $7$ cm; then an equilateral triangle of side $6$ cm.

Solution

Solution of Exercise 33.4.

Isosceles: base $5$ cm, both arcs with the same $7$ cm opening. Equilateral: base $6$ cm, both arcs with the $6$ cm opening.

Exercise 33.5 ★

Construct a square of side $5$ cm. Which tools guarantee the right angles? Which guarantee the equal sides?

Solution

Solution of Exercise 33.5.

The set square guarantees the right angles; the compass (or the ruler) carries the $5$ cm side to all four sides.

Exercise 33.6 ★

Construct a rhombus of side $4$ cm that is not a square (choose your own opening for the first corner). Check the four sides with the compass.

Solution

Solution of Exercise 33.6.

Draw a side $[AB]$ of $4$ cm, choose a direction for $[AD]$ (not perpendicular, to avoid the square), $AD = 4$ cm; then arcs of $4$ cm from $B$ and from $D$ cross at $C$ . The compass confirms the four equal sides.

Exercise 33.7 ★

Draw two points $A$ and $B$ with $AB = 5$ cm. Construct all points that are at $4$ cm from $A$ and at $3$ cm from $B$ . How many are there?

Solution

Solution of Exercise 33.7.

The circle of center $A$ , radius $4$ cm, and the circle of center $B$ , radius $3$ cm, cross in two points (one above, one below the line $(AB)$ ), because $4 + 3 > 5$ and the circles overlap.

Exercise 33.8 ★

Write a construction program (numbered steps, named points) for a rectangle $6$ cm $\times$ $2$ cm topped by an isosceles triangle with equal sides $4$ cm (a pencil shape). Have a classmate execute it.

Solution

Solution of Exercise 33.8.

One correct program: “1. Draw a rectangle $ABCD$ with $AB = 6$ cm (bottom) and $BC = 2$ cm. 2. Draw two arcs of radius $4$ cm centered at $D$ and at $C$ (the top corners); name $S$ their crossing above the rectangle. 3. Draw $[DS]$ and $[CS]$ .” (The point of the pencil is the isosceles triangle $DSC$ .)

Exercise 33.9 ★★

Try to construct a triangle with sides $8$ cm, $3$ cm and $4$ cm. What happens with the arcs? Explain in one sentence why this triangle cannot exist.

Solution

Solution of Exercise 33.9.

The arcs of radii $3$ cm and $4$ cm, drawn from the two ends of the $8$ cm base, never meet: even end to end, $3 + 4 = 7$ cm cannot stretch across $8$ cm. Two sides together must always be longer than the third — otherwise the triangle cannot close.

Exercise 33.10 ★★

Construct a rhombus whose diagonals measure $6$ cm and $4$ cm, knowing that the diagonals of a rhombus cross at their middles at a right angle: draw the diagonals first, then join their four ends.

Solution

Solution of Exercise 33.10.

Draw a segment of $6$ cm and its midpoint $O$ ; draw the perpendicular at $O$ ; mark on it two points at $2$ cm from $O$ (one on each side); join the four endpoints in order. The compass confirms the four equal sides.

Exercise 33.11 ★★

A goat is tied to a stake $A$ by a $4$ m rope, in a flat field with a straight fence passing $3$ m from $A$ . Draw a plan (1 cm per meter): which region can the goat reach? Where does the fence limit it?

Solution

Solution of Exercise 33.11.

On the plan, the goat reaches the disk of center $A$ and radius $4$ cm, cut off by the fence line at $3$ cm from $A$ : her region is the disk minus the sliver beyond the fence. Along the fence she can graze on a segment (where the circle crosses the fence line); beyond it, nothing.