Constructions in geometry involve drawing a geometrical figure using a few construction tools like a ruler, a protractor, set squares and compasses. In mathematical equations, we construct several geometrical figures like triangles, quadrilaterals, circles and arcs, angles, etc.

In this section, we will learn some more geometric constructions. Also, learn how to draw these figures using different construction tools.

What is Geometric Construction?

Constructions using compass and straightedge have a long history in Euclidean geometry. Their use reflects the fundamental axioms of this system. However, the requirement that the only tools used in construction is artificial and only has meaning if the construction of one view is an application of logic.

Construction in geometry is to draw shapes like triangles, quadrilaterals, angles, or lines accurately. These constructions use only a compass, a straightedge (i.e. ruler) and a pencil.

Construction of Triangles

We have seen that a unique triangle can be constructed if out of its six elements (three edges and three angles), any three elements are known. Therefore, for drawing a triangle, three measurements are needed.
To construct a triangle, we should know the properties of triangles.

Triangles are classified based on sides or angles and the following essential properties concerning triangles:

1. The exterior angle of a triangle is equal in the measuring with the sum of interior opposite angles.
2. The total measure of the three angles of a triangle is \(180^{\circ}\).
3. The sum of lengths of any two sides of a triangle is greater than the length of the third side.
4. In every right-angled triangle, the square of hypotenuse length is equal to the sum of the squares of the lengths of the remaining two sides.

Triangle could be drawn if any one of the following sets of elements is given:

1. Three sides.
2. Two sides and one included angle.
3. Two angles and one included side.
4. The hypotenuse and a leg in the right-angled triangle.

To construct a triangle when the lengths of its three sides are known. Here, we would construct triangles when all their sides are known. We draw first a rough sketch to give an idea of where the sides are and then begin by drawing any one of the three lines. See the following example:

Example: Construct a triangle \(ABC\), given that \(AB =5\,\rm{cm}\), \(BC = 6\,\rm{cm}\) and \(AC = 7\,\rm{cm}\).

Solution:
Step 1: Let us draw a rough drawing with given measures.

Step 2: Draw the line segment \(BC\) of measure \(6\,\rm{cm}\).
Step 3: To \(B\), point \(A\) is from a distance of \(5\,\rm{cm}\). So, with \(B\) as the centre, draw an arc of radius \(5\,\rm{cm}\).
Step 4: To \(C\), point \(A\) is from a distance of \(7\,\rm{cm}\). Thus, by \(C\) as the centre, draw an arc of radius \(7\,\rm{cm}\).
Step 5: A must be on both the arcs drawn. Thus, it is the point of intersection of arcs. It is marking the point of intersection of arcs as \(A\). Join \(AB\) and \(AC\). Triangle \(∆ABC\) is now ready.

Construction of Quadrilaterals

A quadrilateral includes ten parts in all: four sides, four angles and two diagonals. To draw a quadrilateral, we will require data about five necessary parts of it.

Construction of Quadrilaterals Basics Rule

1. One unique quadrilateral may be drawn if five magnitudes of a quadrilateral are given. In such a quadrilateral, there exist \(4\) sides, \(4\) angles and \(2\) diagonals. When every four parts of a quadrilateral are well-known, a unique quadrilateral will not be drawn. If five quadrilateral parts are well-known, then a unique quadrilateral may be drawn.
2. Draw a rough drawing of the quadrilateral. Write the five given quadrilateral elements in the diagram. Then, we draw the quadrilateral in order to analyze the given data (measurements).
3. We divide the required quadrilateral in two triangles that could be easily constructed. Those two triangles together will shape a quadrilateral.

Quadrilaterals Properties

1. A quadrilateral can be constructed if the lengths of its four sides and diagonal are given.
2. A quadrilateral can be constructed two diagonals, and three sides are given.
3. A quadrilateral can be constructed if two adjacent sides and three angles are given.
4. A quadrilateral can be constructed if three sides and two included angles are given.

When four sides and one diagonal are given:
A quadrilateral can be constructed if its four sides’ lengths and diagonal are given.
Example: Construct a quadrilateral \(ABCD\) where \(AB = 4\,\rm{cm}\), \(BC = 6\,\rm{cm}\), \(CD = 5\,\rm{cm}\), \(DA = 5.5\,\rm{cm}\) and \(AC = 7\,\rm{cm}\).
Solution: A rough drawing will help us in visualising the quadrilateral.

Step 1: From the rough drawing, it is easy to see that \(∆ABC\) may be constructed using \(SSS\) construction conditions. Draw \(∆ABC\).
Step 2: Now, we must find the fourth point \(D\). This \(D\) would have been on the side opposite to \(B\) with the reference \(AC\). For that, we will have two measurements.
\(D\) is \(5.5\,\rm{cm}\) ahead of \(A\). Thus, with \(A\) as the centre, draw an arc of radius \(5.5\,\rm{cm}\).
Step 3: \(D\) is \(5\,\rm{cm}\) ahead of \(C\). Thus with \(C\) as a centre, draw an arc of radius \(5\,\rm{cm}\).
Step 4: \(D\) must lie on each of the arcs drawn. Thus, it is the point of intersection of the two arcs: Mark \(D\) and completed \(ABCD\). \(ABCD\) is the necessary quadrilateral.

Construction of Parallelogram

Properties of a Parallelogram

1. The opposite edges of a parallelogram are parallel.
2. The opposite edges of a parallelogram are equal.
3. The opposite angles of a parallelogram are equal.
4. The diagonals of a parallelogram bisect to each other.

When two sides and the included angle are given:
Example: To construct a parallelogram \(PQRS\) in such a way that \(PQ = 5.1\,\rm{cm}\), \(QR = 4.5\,\rm{cm}\) and \(∠PQR = 45°\).
Solution:
Steps of construction
Step 1: Draw a rough drawing of the parallelogram \(PQRS.\) Step 2: Draw \(PQ = 5.1\,\rm{cm}\) and at \(Q\), construct a line \(QX\) at an angle of \(45°.\)
Step 3: Cut in \(QR = 4.5\,\rm{cm}\) from \(QX\).
Step 4: Taking \(P\) as a centre, draw an arc of radius \(4.5\,\rm{cm}\) and taking \(R\) as a centre, draw an arc of \(PQRS\) is the necessary parallelogram.

Construction of Rhombus

Properties of Rhombus

1. All the edges of a rhombus are of the same measure.
2. The opposite angles of a rhombus are of the same measure.
3. The adjoining angles of a rhombus are supplementary.
4. The diagonals of a rhombus bisect with one another at right angles.

When one edge and one angle are given:
Example: To construct a rhombus \(PQRS\) in such a way that \(∠P = 45°\) and \(PQ = 3.5\,\rm{cm}\).
Steps of construction
Step 1: Draw a rough drawing of rhombus \(PQRS\)
Step 2: Draw \(PQ = 3.5\,\rm{cm}\). At \(P\), construct \(∠QPX = 45°\) and from \(PX\) cut in \(PS = 3.5\,\rm{cm}\).
Step 3: With \(Q\) as a centre, draw an arc of radius \(3.5\,\rm{cm}\) and from \(S\), draw an arc of radius \(3.5\,\rm{cm}\) meeting the preceding arc at \(R\). Join \(QR\) and \(RS\). \(PQRS\) is the necessary rhombus.

Constructions Tool

To construct geometrical shapes, we need the essential geometrical tools. The few geometrical tools are rulers, compasses, protractors, etc.

Ruler

A ruler is a straight-edged thin strip of steel plastic or some other metal. It has centimetre and millimetre marks on one edge. The marks on the ruler are called graduations, and the ruler is called a graduated ruler. A sample of the ruler is shown in the figure below.

A ruler is used for measuring the line segment or used to draw the line segment.

Compass

A compass has two arms, which are hinged together. One of the arms is equipped with a metal endpoint. The other arm has a screw arrangement that can tightly hold a pencil. The endpoint of the pencil can be adjusted at any distance from the metallic endpoint.

In construction, compasses are used to construct a line segment, angle, perpendicular bisector, and angle bisector etc.

Protractor

A protractor is a geometrical instrument that is used for measuring a given angle. It is also used for constructing an angle of a given magnitude. A protractor is semi-circular in shape and is usually made of plastic or metal sheet. It has degree marks on the curved edge (semi-circular arc) and a \(0 – 180\) line parallel to the straight edge. The mid-point on the protractor is \(90\).

Solved Examples

Q.1. To construct a quadrilateral \(ABCD\) in such a way that \(AB = 2.4\,\rm{cm}\), \(BC = 3.2\,\rm{cm}\), \(AD = 4\,\rm{cm}\), \(∠A = 120°\) and \(∠B = 45°\).
Ans: Steps of construction:
Step 1: Draw a rough drawing of the quadrilateral \(ABCD\).

Step 2: Construct \(AB = 2.4\,\rm{cm}\) and taking \(A\) as a centre, construct \(120°\) and draw a ray \(m\).
Taking \(B\) as centre construct \(45°\) and draw a ray \(n\).
Step 3: Cut in \(AD = 4\,\rm{cm}\) on ray \(m\) and \(BC = 3.2\,\rm{cm}\) on ray \(n\). Join \(CD\).
\(ABCD\) is the necessary quadrilateral.

Q.2. To construct a parallelogram \(ABCD\) in such a way that \(AB = 6\,\rm{cm}\), \(BC = 5.5\,\rm{cm}\) and \(BD = 6.2\,\rm{cm}.\)
Ans: Steps of constructions:
Step 1: Draw a rough drawing of the parallelogram \(ABCD\).

Step 2: Construct \(AB = 6\,\rm{cm}\). Taking \(B\) as a centre, draw an arc of radius \(6.2\,\rm{cm}\) and taking \(A\) as a centre, draw an arc measuring \(5.5\,\rm{cm}\) in such a way that it cuts the preceding arc at \(D\). Join \(AD\) and \(BD\).
Step 3: Taking \(D\) as a centre, draw an arc of radius \(6\,\rm{cm}\) and taking \(B\) as a centre, draw an arc of radius \(5.5\,\rm{cm}\) cutting the preceding arc at \(C\). Join \(BC\) and \(CD\). \(ABCD\) is the necessary parallelogram.

Q.3. Construct an equilateral \(∆PQR\) with sides \(5\,\rm{cm}\).
Ans: Steps of construction:
Step 1: Draw a line segment \(QR = 5\,\rm{cm}\).
Step 2: Keep the compass at point \(Q\), and draw an arc with a radius \(5\,\rm{cm}\) by using a compass.
Step 3: Keep the compass at point \(R\), and draw an arc with a radius \(5\,\rm{cm}\) by using a compass.
Step 4: Mark the intersection points as \(P\). Now, join \(PQ\) and \(PR\).

Q.4. To construct a rhombus \(PQRS\) such that \(PR = 5.2\,\rm{cm}\) and \(QS = 7.4\,\rm{cm}\).
Ans: Steps of construction:
Step 1: Draw a rough drawing of the rhombus \(PQRS\).

Step 2: Draw \(QS = 7.4\,\rm{cm}\) and draw its perpendicular bisector \(XY\) cutting it at \(O\).
Step 3: From \(O\), cut \(OP = OR = 2.6\,\rm{cm}\). Join \(PQ,\,QR,\,RS\) and \(PS\).
\(PQRS\) is the necessary rhombus.

Q.5. To construct a quadrilateral \(PQRS\) such that \(PQ = 3.5\,\rm{cm}\), \(QR = 2.1\,\rm{cm}\), \(PS = 1.9\,\rm{cm}\), \(QS = 3.1\,\rm{cm}\) and \(∠PQR = 60°\).
Ans: Steps of construction:
Step 1: Draw a rough drawing of the quadrilateral \(PQRS\).

Step 2: Construct \(PQ = 3.5\,\rm{cm}\). At \(Q\), draw \(60°\) and draw a ray \(m\). Cut off \(QR = 2.1\,\rm{cm}\) from it.
Step 3: Taking \(Q\) as a centre, draw an arc of radius \(3.1\,\rm{cm}\) and by taking \(P\) as a centre, draw an arc of radius \(1.9\,\rm{cm}\) cutting the preceding arc at \(S\). Join \(PS,\,QS\) and \(SR\).
\(PQRS\) is the necessary quadrilateral.

Summary

In this article, we learned about the essential tools used in constructing geometrical figures, triangles, quadrilaterals, parallelograms, and rhombus. We looked at a few solved examples on constructions and FAQs on constructions.

This article’s learning outcome is that we understood how to construct basic geometric shapes that give an idea to design architecture and help in the engineering field.

Learn Important Geometry Formulas

FAQs

Q.1. What are the tools used in geometric constructions?
Ans: To construct geometrical shapes, we need the basic geometrical tool. The few geometrical tools are ruler, compasses, protractor etc.

Q.2. What are the different types of construction in geometry?
Ans: The few different types of construction in geometry are:
1. Construction of bisectors of lines and angles
2. Construction of circles
3. Construction of a tangent to a circle
4. Construction of polygons

Q.3. What is straightedge geometry?
Ans: A straightedge is a tool used for drawing line segments or measuring the length of a line segment; it is commonly called a ruler.

Q.4. How vital is geometric construction in your daily life?
Ans: Geometric constructions help us to design houses, buildings, and monuments with proper ventilation.

Q.5. What is the purpose of geometric constructions?
Ans: Geometric constructions help us to draw lines, angles, and shapes with simple tools.