Constructing Quadrilaterals

In geometry, a quadrilateral is a figure with four sides and four angles. Quadrilaterals are very common shapes that we see everywhere around us, from books to tables, windows to kites. Understanding how to draw a quadrilateral can help us solve more complex mathematical problems in daily life.

Basic elements to draw a quadrilateral

Before we dive into the construction process, let's get acquainted with some of the basic elements required to build a quadrilateral:

• Sides: A quadrilateral has four sides or edges.
• Angles: A quadrilateral has four angles.
• Vertices: The points where two sides meet are called vertices. A quadrilateral has four vertices.
• Diagonal: The line segment joining two opposite vertices is called diagonal. A quadrilateral has two diagonals.

Types of quadrilaterals

Quadrilaterals can be classified into different types based on their sides and angles. Here are some common types:

• Square: All sides are equal, and all angles are 90 degrees.
• Rectangle: Opposite sides are equal and parallel, and all angles are 90 degrees.
• Rhombus: All sides are equal, and opposite angles are also equal. Diagonals bisect each other at right angles.
• Parallelogram: Opposite sides are equal and parallel. Opposite angles are equal.
• Trapezium (or trapezium): only one pair of opposite sides are parallel.
• Kite: Two pairs of adjacent sides are equal, and one pair of opposite angles are equal.

Methods for constructing a quadrilateral

The construction of a quadrilateral depends on the given information, such as the length of the side, the angles, or the length of the diagonal. Let's explore some common methods:

1. Construction of a quadrilateral given four sides and one angle

Suppose we want to construct a quadrilateral whose sides have lengths AB, BC, CD, DA and angle ABC.

1. Draw the first side AB according to the given length.
2. Using the protractor at point B draw an angle (angle ABC) with the given measure.
3. From point B, draw another side BC of the given length, making sure it aligns with the angle drawn in step 2.
4. From point C, draw the third side CD according to the given length.
5. Finally, join point D to point A to form the last side of the quadrilateral.

You will now have a complete quadrilateral ABCD.

2. Constructing a quadrilateral when given two diagonals and three sides

Sometimes, two diagonals and three sides may be enough information. Suppose we are given the diagonals AC and BD, and the sides AB, BC, and CD.

1. First draw a diagonal AC of the correct length.
2. From point A, draw side AB at the correct length.
3. Draw side CD from point C to the correct measurement.
4. Join B and D, making sure both are on opposite sides of AC. Use a compass to measure the given diagonal BD.
5. Finally, join B to C. Now, you have a quadrilateral ABCD.

3. Construction of special quadrilaterals

Building a square

A square is a special type of quadrilateral in which all four sides are equal and all angles are right angles.

1. Draw a line the length of the desired side.
2. Use the protractor to measure and draw a 90 degree angle at one end of your line.
3. Using a compass, measure the same length on this new line and mark it.
4. Repeat the process to create a square from this new point.
5. Connect the end of your new line to the opposite end of your starting line.

Your ideal square is ready.

Constructing a rectangle

The opposite sides of a rectangle are equal and parallel, and all angles are 90 degrees.

1. Draw a line segment for one side of the rectangle.
2. At each end point, use the protractor to draw a 90 degree angle.
3. Measure the length of the opposite side using a compass and mark the end points.
4. Join these new endpoints to form a complete rectangle.

Construction of rhombus

All the sides of a rhombus are equal, and its opposite sides are parallel.

1. Draw one side of the rhombus to the desired length.
2. Measure an interior angle using a protractor.
3. Use the measured angle to draw an adjacent side of the same length.
4. Repeat the process for the remaining sides.
5. Draw a rhombus by joining all the endpoints.

Key takeaways

• The choice of construction method depends on the given information (sides, angles, diagonals).
• Use a protractor to measure angles and a compass to measure lengths.
• Always make sure your drawings and construction are clean and clear.

Practice exercises

1. Construct a quadrilateral with sides AB = 5 cm, BC = 4 cm, CD = 6 cm, DA = 3 cm and ∠B = 70°.
2. Construct a rectangle in which one side is 6 cm and the adjacent side is 4 cm. Verify that all the angles are 90°.
3. Construct a rhombus with side 5 cm and one angle of 60°.

Enjoy building quadrilaterals and exploring the fascinating world of geometry!

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