---
title: "Geometric Constructions"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 33
exercises: 11
source: https://one-course.com/books/math/1/en/chapter/33-geometric-constructions
---

# Chapter 33 — Geometric 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](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-def), [parallels](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-def), triangles and the special [quadrilaterals](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon) — craft first, theory later (Chapters [40](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#ch-g6-lines) and [41](https://one-course.com/books/math/1/en/chapter/41-triangles-and-quadrilaterals#ch-g6-shapes)).

## 33.1 Perpendiculars and parallels through a point

**Method 33.1 (The two basic constructions).**

*[Perpendicular](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-def) to $d$ through $A$*: slide the set square along $d$ until its other right-angle edge reaches $A$; draw; mark the [right angle](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle).

*[Parallel](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-def) 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](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-def) at $A$; [perpendicular](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-def) 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:

1. draw the longest side, $[AB]$ with $AB = 6$ cm;
2. arc of center $A$ and [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $5$ cm;
3. arc of center $B$ and [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $4$ cm;
4. their crossing is the third [vertex](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon) $C$ ; 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 .)](https://one-course.com/images/onecourse/chapters/math-1/g5-geometry/fig-b167d3504da4.svg)

*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](https://one-course.com/books/math/1/en/chapter/51-triangles-and-angles#ch-g7-triangles).)*

**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).**

1. *Rectangle* : two pairs of [parallels](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-def) crossing at [right angles](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle) — or the chain of [Example 33.2](#ex-g5-geometry-chain) ;
2. *square* : a rectangle whose four sides use the same length — carry it with the compass;
3. *rhombus* : four equal sides — draw one side, then two compass arcs from its ends with the same opening fix the other two [vertices](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon) (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](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon) with four $4$ cm sides and one [right angle](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle)” — build it: the four equal sides force a rhombus shape, and pressing one corner to a [right angle](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle) straightens all the others: the result is a square. Some descriptions secretly describe a more special shape than they announce. (Why the [right angle](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle) spreads to all four corners is explained in [Chapter 52](https://one-course.com/books/math/1/en/chapter/52-central-symmetry-and-parallelograms#ch-g7-central).)

## 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](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $3$ cm. “Find all points at $3$ cm from $A$ *and* $4$ cm from $B$”: draw both circles; their crossings (two, one or [zero](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-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](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-def) to $d$ through $A$, then the [parallel](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-def) to $d$ through $A$. Mark the [right angles](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle).

**Solution of Exercise 33.1.**

Both constructions follow [Method 33.1](#met-g5-geometry-basic); the [perpendicular](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-def) and the [parallel](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-def) through $A$ are themselves [perpendicular](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-def) to each other.

**Exercise 33.2 ★.**

Construct a rectangle $ABCD$ with $AB = 7$ cm and $BC = 4$ cm, following [Example 33.2](#ex-g5-geometry-chain). Check the fourth angle with the set square: if the construction is exact, it is right automatically.

**Solution of Exercise 33.2.**

Chain of [perpendiculars](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-def) as in [Example 33.2](#ex-g5-geometry-chain). 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](#met-g5-geometry-triangle)).

**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 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](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle)? Which guarantee the equal sides?

**Solution of Exercise 33.5.**

The set square guarantees the [right angles](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle); 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 of Exercise 33.6.**

Draw a side $[AB]$ of $4$ cm, choose a direction for $[AD]$ (not [perpendicular](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-def), 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 of Exercise 33.7.**

The circle of center $A$, [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $4$ cm, and the circle of center $B$, [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $3$ 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 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](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $4$ 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 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](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle): draw the diagonals *first*, then join their four ends.

**Solution of Exercise 33.10.**

Draw a segment of $6$ cm and its midpoint $O$; draw the [perpendicular](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-def) 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 of Exercise 33.11.**

On the plan, the goat reaches the disk of center $A$ and [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $4$ 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.