Construction of Quadrilaterals

Quadrilateral Construction and Types:

A plane closed geometric figure with four sides and four angles is called a quadrilateral. All the shapes with four sides and  four vertices are said to be quadrilaterals. The different types of quadrilaterals are Kite, parallelogram, trapezium, kite, rectangle, rhombus, and square. All these quadrilaterals have their own properties. A quadrilateral can be constructed using a ruler and compass only without using any other devices in the instrument box.

Properties of Quadrilaterals:

  • A quadrilateral has four sides and four angles.

  • The sides may be of the same length or different lengths.

  • The sum of all the four interior angles of a quadrilateral is equal to 3600. This is called the angle sum property of a quadrilateral.

  • The sum of all the four exterior angles of a quadrilateral is equal to 3600.

Construction of Quadrilaterals:

Quadrilaterals can be constructed with the help of a ruler and compass under any one of the following quadrilateral constructions and types.

Case 1: Length of 4 sides and one diagonal are given

Case 2: Measurement of three sides and two angles are given

Case 3: Measurement of 2 sides and 3 angles are given

It is to be remembered that the quadrilaterals are generally labeled with the uppercase English alphabets that are consecutive. 

Problems on Construction of Quadrilaterals when 4 sides and one diagonal are given:

When the length of four sides and a diagonal are given, the following steps are followed to construct the quadrilateral. 

Example: Construct a quadrilateral PQRS where PQ = 5cm, QR = 3 cm, RS = 5 cm, SP = 4 cm and SQ = 6 cm.


Step 1: Draw a rough diagram of the quadrilateral to be constructed. 

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Step 2: Draw the line of the longest side as the base using the ruler. 


Construct a base of 5 cm and label it as RS.


Step 3: Set the compass to the radius measuring the side of the quadrilateral and cut an arc from the vertex of the base. Now set the compass to the radius equal to the length of the diagonal and cut an arc from the opposite vertex. Label the vertices accordingly.

Set the compass radius to 3 cm and cut an arc from R. Set the radius equal to 6 cm and cut an arc from S. The point of intersection of both the arcs is the vertex ‘Q’. Join RQ and SQ.

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Step 4: Cut an arc from the appropriate vertices measuring the lengths of the other two sides of the quadrilateral. 


From the vertex ‘Q’ cut an arc of length 5 cm and from the vertex ‘S’ cut an arc of radius 4 cm. The point of intersection of these two arcs is the vertex ‘P’. Join PS and PQ.

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This gives the complete construction of quadrilateral PQRS. 

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Problems on Construction of Quadrilateral when 3 Sides and Two Angles are Given:

Example: Construct the quadrilateral PQRS given that the sides QR = 6 cm, RS = 5 cm and PS = 4cm and ㄥS = 1000 and ㄥR = 1200.

Solution:


Step 1: Draw the rough diagram of quadrilateral PQRS.

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Step 2: Construct a line segment RS = 5 cm. 

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Step 3: Construct the angles 1000 and 1200 at S and R respectively.

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Step 4: Cut an arc of 4 cm from S and 6 cm from R on the line of constructed angles. 

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Step 5: Label the point of intersections as P and Q. Join PQ.

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Construction of Quadrilaterals when Two Sides and 3 Angles are Given:

Example: Construct the quadrilateral ABCD in which AB = 5 cm and BC = 3 cm. The angles ㄥA = 1200, ㄥB = 1100 and ㄥC = 900.

Solution:

Step 1: Draw a rough figure of the quadrilateral

Step 2: Construct a line segment AB of length 5 cm.

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Step 3: From point A, construct an angle = 1200 and from point B construct an angle = 1100.


Step 4: Cut an arc of 3 cm on the line drawn at B. The point of intersection of the line and the arc is the vertex C. 

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Step 5: Construct an angle = 900 at the vertex C. Extend the line to meet the line drawn at A.

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Step 6: The point of intersection of both the lines is the vertex D. 

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Fun Facts:

  • All the sides and vertices of a quadrilateral lie on the same plane. A quadrilateral is a 2-dimensional geometrical shape. It is a four-sided closed polygon.

  • From the problems of construction of quadrilaterals, all parallelograms and squares are quadrilaterals. But, all the quadrilaterals are squares or parallelograms.

  • For the quadrilateral construction and types, any 5 of the 10 dimensions are required. The 10 dimensions are 2 diagonals, 4 sides, and 4 angles.

FAQ (Frequently Asked Questions)

1. What is a Quadrilateral? What are the Types of Quadrilaterals?

A quadrilateral is a 2-dimensional closed geometrical shape with 4 sides and 4 angles. The different types of quadrilaterals are:

  • Trapezium: One pair of opposite sides are parallel. All parallelograms, squares, and rectangles are trapezoids.

  • Parallelograms: If in a quadrilateral, both the pairs of opposite sides are parallel and equal. Opposite angles are equal.

  • A rectangle is a special type of parallelogram in which all the angles are equal to 900.

  • A square is a special type of rectangle in which all the sides are equal.

  • A rhombus is a parallelogram in which all sides are equal and opposites sides are parallel.

  • A kite is a quadrilateral in which two pairs of adjacent sides are equal and a pair of opposite angles are equal. 

2. How are Problems with the Construction of Quadrilaterals Solved?

Quadrilaterals can be constructed only with the help of a ruler and compass. However, the basic knowledge of construction of angles without using a protractor is necessary. The quadrilateral has 4 sides, 4 angles, and 2 diagonals. If any of the 5 quantities among the 10 are given, the quadrilateral can be constructed. Three cases under which a quadrilateral can be constructed are:

  • When 4 sides and one diagonal are given

  • When 3 sides and 2 angles are given

  • When 2 sides and 3 angels are given