Construction of Quadrilaterals: Definition, Rules, Examples

Suppose any three parts are known out of its six parts, a unique triangle. Some of the measurements of a quadrilateral must be known for the construction of quadrilaterals. As the quadrilateral has four sides, four angles and two diagonals

Four measurements are not sufficient to draw a unique quadrilateral. To draw a unique quadrilateral, we need \(5\) measures of the quadrilateral. Therefore, a quadrilateral can be drawn if five measurements of the quadrilateral are known. This article shall discuss how to draw a quadrilateral with examples.

Define Construction of Quadrilaterals

Since three measurements are enough to draw a triangle, a natural question arises whether four measures would be sufficient to draw a unique four-sided closed figure, a quadrilateral. 

The answer is no; a quadrilateral has ten parts: four sides, four angles and two diagonals. Therefore, to construct a quadrilateral, we will need data of a minimum of five elements.

Construction of Quadrilaterals Basics Rule

1. A unique quadrilateral can be drawn if five measurements of a quadrilateral are given. In a quadrilateral, there are \(4\) sides, \(4\) angles and \(2\) diagonals. A unique quadrilateral cannot be drawn when any four parts of a quadrilateral are known. But, if five parts of a quadrilateral are known, then a unique quadrilateral can be drawn.
2. First, we draw a rough sketch of the quadrilateral and write the measurements of the five parts of the quadrilateral in the rough figure. Then, we draw the quadrilateral analysing the given data (measurements).
3. We divide the necessary quadrilateral into two triangles which can be easily constructed. These two triangles together will form a quadrilateral.

Construction of Quadrilaterals – Combination of Elements

1. A quadrilateral can be constructed uniquely if the lengths of its four sides and diagonal are given.
2. A quadrilateral can be constructed uniquely if its two diagonals and three sides are given.
3. A quadrilateral can be constructed in a unique way if its two adjacent sides and three angles are given.
4. A quadrilateral can be constructed in a unique way if its three sides and two included angles are given.
5. A quadrilateral can be constructed when four sides and one angle are given.

Notes

A quadrilateral can be constructed using a ruler and compass. In the construction of quadrilaterals, we use a compass to construct angles and to mark arcs to draw line segments.

When four sides and one diagonal are given

Example: Construct a quadrilateral \(PQRS\) in which \(P Q=3 \mathrm{~cm}, Q R=5 \mathrm{~cm}, Q S=5 \mathrm{~cm}, P S=4 \mathrm{~cm}\) and \(R S=4 \mathrm{~cm}\).

Ans:

Thus, quadrilateral \(PQRS\) will be formed.

Steps of construction:

1. Draw a line segment \(P Q=3 \mathrm{~cm}\).
2. Taking \(P\) as the centre, draw an arc of radius \(P S=4 \mathrm{~cm}\).
3. Taking \(Q\) as a centre, draw an arc of radius \(Q S=5 \mathrm{~cm}\), which cuts the arc of step \(2\) at \(S\).
4. Taking \(S\) as a centre, draw an arc of radius \(R S=4 \mathrm{~cm}\).
5. Taking \(Q\) as a centre, draw an arc of radius \(Q R=5 \mathrm{~cm}\), which cuts the arc drawn in step-\(4\) at \(R\).
6. Join \(P S, Q S, Q R\) and \(R S\).
   Thus \(P Q R S\) is the required quadrilateral formed.

When two diagonal and three sides are given

Example: Construct a quadrilateral \(PQRS\) in which \(Q R=7.5 \mathrm{~cm}, P R=P S=6 \mathrm{~cm}, R S=5 \mathrm{~cm}\) and \(Q S=10 \mathrm{~cm}\).

Ans:

Thus quadrilateral \(PQRS\) will be formed.

Steps of construction:

1. Draw a line segment \(Q R=7.5 \mathrm{~cm}\).
2. Taking \(Q\) as a centre and draw an arc of radius \(Q S=10 \mathrm{~cm}\).
3. Taking \(R\) as a centre, draw an arc of radius \(R S=5 \mathrm{~cm}\), which cuts the arc drawn in step \(-2\) at \(S\).
4. Taking \(S\) as the centre, draw an arc of radius \(S P=6 \mathrm{~cm}\).
5. Taking \(R\) as the centre, draw an arc of radius \(R P=6 \mathrm{~cm}\), which cuts the arc drawn in step \(-4\) at \(P\).
6. Join \(Q S, R S, P S, P R\) and \(P Q\).
   \(P Q R S\) is the required quadrilateral formed.

Solved Examples

Q.1. Construct a parallelogram \(P Q R S\) in which \(P Q=6 \mathrm{~cm}, Q R=4.5 \mathrm{~cm}\) and diagonal \(P R=6.8 \mathrm{~cm}\).
Ans: Draw a rough sketch of the required parallelogram and write down the given dimensions.

1. Draw \(P Q=6 \mathrm{~cm}\)
2. With \(P\) as centre and radius \(6.8 \mathrm{~cm}\), draw an arc.
3. With \(Q\) as centre and radius, \(4.5 \mathrm{~cm}\) draw another arc, cutting the previous arc at \(R\)
4. Join \(QR\) and \(PR\).
5. With \(P\) as centre and radius \(4.5 \mathrm{~cm}\), draw an arc.
6. With \(R\) as centre and radius, \(6 \mathrm{~cm}\) draw another arc, cutting the previously drawn arc at \(D\).
7. Join \(SP\) and \(SR\).
   Then, \(P Q R S\) is the required parallelogram.

Q.2. Construct a quadrilateral \(P Q R S\) in which \(P Q=3.5 \mathrm{~cm}, Q R=6.5 \mathrm{~cm}\) and \(\angle P=\angle R=105^{\circ}\) and \(\angle S=75^{\circ}\).
Ans: From the rough sketch, it is clear that quadrilateral \(PQRS\) will be formed only when \(\angle Q\) is known. For this, we know that
\(\angle P+\angle Q+\angle R+\angle S=360^{\circ}\)
\(\Rightarrow 105^{\circ}+\angle Q+105^{\circ}+75^{\circ}=360^{\circ}\)
\(\Rightarrow \angle Q=75^{\circ}\)

Steps of construction:

1. Draw a line segment \(Q R=6.5 \mathrm{~cm}\).
2. At \(Q\), make \(\angle R Q A=75^{\circ}\)
3. From \(Q A\), cut \(Q P=3.5 \mathrm{~cm}\)
4. At \(R\), make \(\angle Q R B=105^{\circ}\)
5. At \(P\), make \(\angle Q P S=105^{\circ}\)
   Ray \(P C\) meets ray \(R B\) at \(S\).
   \(P Q R S\) is the required quadrilateral formed.

Q.3. Construct a quadrilateral \(PQRS\) in which \(PQ = 5.5~{\text{cm}},\,QR = 3.7~{\text{cm}},\,\angle P = 60^\circ ,\,\angle Q = 105^\circ\) and \(\angle R = 90^\circ \).
Ans:

Steps of construction: 

1. Draw a line segment \( Q=5.5 \mathrm{~cm}\).
2. At \(P\), make \(\angle Q P Y=60^{\circ}\).
3. At \(Q\), make \(\angle P Q R=105^{\circ}\).
4. Taking \(Q\) as the centre, draw an arc of radius \(Q R=3.7 \mathrm{~cm}\), which cuts \(Q Z\) at \(R\).
5. At \(R\), make \(\angle Q R X=90^{\circ}\). Ray \(R X\) cuts ray \(P Y\) at \(S\).
   \(P Q R S\) is the required quadrilateral formed.

Q.4. Construct a quadrilateral \(PQRS\) in which \(P Q=4.2 \mathrm{~cm}, Q R=3.6 \mathrm{~cm}, R S=4.8 \mathrm{~cm}, \angle Q=30^{\circ}\) and \(\angle R = 150^\circ \).
Ans: On drawing a rough sketch, it is clear that any of the triangles \(PQR\) and \(QRS\) can be first made because in both the triangles, two sides and included angle are known.
Thus quadrilateral \(PQRS\) is formed.

Steps of construction:

1. Take a line segment \(Q R=3.6 \mathrm{~cm}\).
2. At \(Q\), make \(\angle R Q Y=30^{\circ}\) and at \(R\) make \(\angle Q R Z=150^{\circ}\).
3. Taking \(Q\) as centre, draw an arc \(P Q=4.2 \mathrm{~cm}\) which cuts \(Q Y\) at \(P\).
4. Taking \(R\) as centre cut \(R S=4.8 \mathrm{~cm}\) from \(R Z\).
5. Join \(P S\).
   \(P Q R S\) is the required quadrilateral formed.

Q.5. Construct a quadrilateral \(PQRS\) in which \(P Q=2.8 \mathrm{~cm}, Q R=3.1 \mathrm{~cm}, R S=2.6 \mathrm{~cm}, S P=3.3 \mathrm{~cm}\) and \(\angle P = 60^\circ \).
Ans:

Steps of construction:

1. Draw a line segment \(P Q=2.8 \mathrm{~cm}\).
2. At \(P\), make \(\angle Q P Y=60^{\circ}\).
3. Taking \(P\) as the centre, draw an arc of radius \(P S=3.3 \mathrm{~cm}\), which cuts \(P Y\) at \(S\).
4. Taking \(S\) as the centre, draw an arc of radius \(S R=2.6 \mathrm{~cm}\).
5. Taking \(Q\) as the centre, draw an arc of radius \(Q R=3.1 \mathrm{~cm}\), which cuts the arc drawn in step \(-4\) at \(R\).
6. Join \(Q R\) and \(R S\).
   \(P Q R S\) is the required quadrilateral formed.

Summary

In this article, we learnt the definition of Construction of Quadrilaterals, Basic Rules of Construction of Quadrilaterals, Construction of Quadrilaterals notes, Solved examples, Frequently Asked Questions.

Learning Outcome: Method to construct a unique quadrilateral from the given measurement.

FAQs

Q.1. In which case can we construct a quadrilateral?
Ans: A quadrilateral can be constructed if the following measurements are given:
1. A quadrilateral can be constructed in a unique way if the lengths of its four sides and a diagonal is given. 
2. A quadrilateral can be constructed in a unique way if its two diagonals and three sides are given. 
3. A quadrilateral can be constructed in a unique way if its two adjacent sides and three angles are given. 
4. A quadrilateral can be constructed in a unique way if its three sides and two included angles are given.

Q.2. How can we construct a quadrilateral?
Ans: A unique quadrilateral can be drawn if five measurements of a quadrilateral are given. First of all, we draw a rough sketch of the quadrilateral and write the measurements of the five given parts of the quadrilateral in the rough figure. Then, we draw the quadrilateral analysing the given data (measurements).

Q.3. How do you construct a quadrilateral with a compass?
Ans: A quadrilateral can be constructed using a ruler and compass. In the construction of quadrilaterals, we use a compass to construct angles and to mark arcs to draw line segments.

Q.4. How do you construct a quadrilateral with three sides and two included angles?
Ans:  To construct a quadrilateral \(PQRS\) in which \(P Q=4.2 \mathrm{~cm}, Q R=3.6 \mathrm{~cm}, \mathrm{RS}=4.8 \mathrm{~cm}, \angle Q=30^{\circ}\) and \(\angle R = 150^\circ \)
Steps of construction:
Take a line segment \(Q R=3.6 \mathrm{~cm}\).
At \(Q\), make \(\angle R Q Y=30^{\circ}\) and at \(R\) make \(\angle Q R Z=150^{\circ}\).
Taking \(Q\) as centre, draw an arc \(P Q=4.2 \mathrm{~cm}\) which cuts \(Q Y\) at \(P\).
Taking \(R\) as centre cut \(R S=4.8 \mathrm{~cm}\) from \(R Z\).
Join \(P S\).
\(P Q R S\) is the required quadrilateral formed.

Q.5. How do you construct a unique quadrilateral?
Ans: A unique quadrilateral can be drawn if five measurements of a quadrilateral are given. In a quadrilateral, there are \(4\) sides, \(4\) angles and \(2\) diagonals. When any four parts of a quadrilateral are known, unique quadrilateral cannot be drawn. But if five parts of a quadrilateral are known, then a unique quadrilateral can be drawn.