ICSE Class 9 Maths Chapter 15 Construction of Polygons

Construction Of Polygons

[Using Ruler and Compass Only]

15.1 Construction Of Quadrilaterals

To construct a quadrilateral means: to find (locate) its four vertices.

Always, draw a rough free-hand sketch, before starting the actual construction.

1. To Construct A Quadrilateral, Whose Four Sides And One Angle Are Given.

Let quadrilateral ABCD has AB = 3-5 cm, BC = 4-0 cm, CD = 5-0 cm, DA = 4-0 cm and angle B = 45 degrees.

Steps:

1. Draw BC = 4-0 cm.

2. Through B, draw BP such that angle B = 45 degrees.

3. From BP, cut BA = 3-5 cm.

4. With A and C as centres and radii 4 cm and 5 cm respectively, draw arcs cutting each other at D.

5. Join AD and CD.

ABCD is the required quadrilateral.

2. To Construct A Quadrilateral, Whose Three Sides And Two Consecutive Angles Are Given.

Let quadrilateral ABCD has AB = 4-0 cm, BC = 4-5 cm, CD = 4-7 cm, angle B = 60 degrees and angle A = 120 degrees.

Steps:

1. Draw BC = 4-5 cm

2. Construct angle MBC = 60 degrees and then from BM cut BA = 4-0 cm.

3. Draw AP such that angle A = 120 degrees.

4. With C as centre and radius = 4-7 cm draw an arc cutting AP at D.

5. Join C and D.

ABCD is the required quadrilateral.

3. To Construct A Quadrilateral, Whose Four Sides And One Diagonal Are Given.

Let quadrilateral ABCD has AB = 3-0 cm, BC = 4-0 cm, CD = 5-0 cm, DA = 3-5 cm and diagonal AC = 4-5 cm.

Steps:

Using given dimensions; first of all construct triangle ABC and then triangle ADC.

4. To Construct A Quadrilateral, Whose Three Sides And Two Diagonals Are Given.

Let quadrilateral ABCD has AB = 3-5 cm, BC = 3-0 cm, AD = 3-8 cm, diagonal AC = 5-0 cm and diagonal BD = 4-5 cm

Steps:

(i) First of all, construct triangle ABC and then triangle ABD.

(ii) Join C and D.

ABCD is the required quadrilateral.

15.2 Construction Of Parallelograms

1. To Construct A Parallelogram, Whose Two Consecutive Sides And The Included Angle Are Given.

Let parallelogram ABCD has AB = 3-0 cm, BC = 4-0 cm and angle B = 60 degrees.

Since, opposite sides of a parallelogram are equal, therefore, AB = DC = 3-0 cm and BC = AD = 4-0 cm.

Steps:

(i) Taking AB = 3 cm, BC = 4 cm and angle B = 60 degrees, construct triangle ABC.

(ii) Now, construct triangle ADC.

ABCD is the required parallelogram.

2. To Construct A Parallelogram, Whose One Side And Both The Diagonals Are Given.

Let parallelogram ABCD has side BC = 4-5 cm, diagonal AC = 5-6 cm and diagonal BD = 5-0 cm.

Steps:

(i) Since diagonals of a parallelogram bisect each other; construct triangle OBC, such that:

OB = \(\frac{1}{2}\) BD = \(\frac{1}{2}\) times 5-0 cm = 2-5 cm

OC = \(\frac{1}{2}\) AC = \(\frac{1}{2}\) times 5-6 cm = 2-8 cm

and, BC = 4-5 cm

(ii) Produce BO upto D, such that OD = OB = 2-5 cm and produce CO upto A, such that OA = OC = 2-8 cm.

(iii) Join AB, AD and CD.

ABCD is the required parallelogram.

3. To Construct A Parallelogram, Whose Two Consecutive Sides And One Diagonal Are Given.

Let parallelogram ABCD has AB = 3-5 cm, BC = 4-5 cm and diagonal BD = 5-5 cm.

Steps:

1. Draw triangle BCD.

2. Draw triangle BAD.

4. To Construct A Parallelogram, Whose Two Diagonals And Included Angle Are Given.

Let parallelogram ABCD has diagonal AC = 5-4 cm, diagonal BD = 4-8 cm and the acute angle between the diagonals = 60 degrees.

Steps:

1. Draw AC = 5-4 cm and locate its mid point O.

2. Draw line BOD such that angle DOC = 60 degrees and OB = OD = \(\frac{1}{2}\) BD = \(\frac{1}{2}\) times 4-8 cm = 2-4 cm.

3. Join AB, BC, CD and DA.

ABCD is the required parallelogram.

In this construction, we are given diagonal AC = 5-4 cm and diagonal BD = 4-8 cm. The lengths of both the diagonals (5-4 cm and 4-8 cm) are divisible by 2, correct to one place of decimal to get 2-7 cm and 2-4 cm respectively. Lengths 2-7 cm and 2-4 cm can easily be measured with the help of a scale in your geometry box.

But, if we are given AC = 5-4 cm and BD = 4-5 cm, now \(\frac{1}{2}\) BD = \(\frac{1}{2}\) times 4-5 cm = 2-25 cm, which can not correctly be measured with the help of the scale you use. To overcome this situation, instead of starting with AC = 5-4 cm, you start the construction of the parallelogram with BD = 4-5 cm and continue accordingly.

Steps:

1. Draw BD = 4-5 cm.

2. Draw perpendicular bisector of BD to get its mid-point O.

3. Draw line AOC, such that angle AOD = 60 degrees.

4. From AOC, cut OA = OC = \(\frac{1}{2}\) AC = \(\frac{1}{2}\) times 5-4 cm = 2-7 cm.

5. Join AB, BC, CD and DA.

ABCD is the required parallelogram.

5. To Construct A Parallelogram, Whose Two Adjacent Sides And Height Are Given.

Let parallelogram ABCD has adjacent sides AB = 3-6 cm, BC = 4-6 cm and height corresponding to side BC = 2-6 cm.

Steps:

1. Draw BC = 4-6 cm.

2. At C, draw CP perpendicular to BC.

3. From CP, cut CE = 2-6 cm = height of parallelogram.

4. Through E, draw perpendicular to CP to get QR parallel to BC.

5. With B as centre and radius = AB = 3-6 cm, draw an arc which cuts QR at A.

6. With C as centre and radius = 3-6 cm, draw one more arc which cuts QR at D.

ABCD is the required parallelogram.

15.3 Construction Of Trapezium

To Construct A Trapezium ABCD, Whose Four Sides Are Given.

Let AD // BC, AD = 3-0 cm, AB = 2-5 cm, BC = 5-0 cm and CD = 2-8 cm.

Steps:

1. Draw BC = 5-0 cm.

2. From BC, cut BE = AD = 3-0 cm.

3. Draw triangle DEC, such that DE = AB = 2-5 cm and CD = 2-8 cm.

4. Taking B and D as centres and radii 2-5 cm and 3-0 cm respectively, draw arcs cutting each other at A.

5. Join AB and AD.

ABCD is the required trapezium.

Teacher's Note

Trapeziums appear in architecture when designing roof structures or decorative panels that require one pair of parallel sides.

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