Constructing Special Quadrilaterals

Class 8Practical Geometry

In Class 8 Practical Geometry, you learn to construct quadrilaterals when certain measurements are given. Special quadrilaterals — square, rectangle, parallelogram, and rhombus — have specific properties that make their construction simpler than constructing a general quadrilateral.

A general quadrilateral requires 5 independent measurements (sides, diagonals, angles) to be uniquely constructed. But special quadrilaterals have constraints (equal sides, right angles, parallel sides) that reduce the number of measurements needed.

All constructions use a ruler (straightedge) and compass only. A protractor may be used for measuring angles where required. The key is to use the defining properties of each special quadrilateral to plan the construction steps.

This topic builds on your knowledge of properties of special quadrilaterals and triangle construction from earlier chapters.

What is Constructing Special Quadrilaterals?

Definition: Constructing a special quadrilateral means drawing it accurately using a ruler and compass, given sufficient measurements, by applying the specific properties of that quadrilateral type.

Properties used in construction:

Square: All 4 sides equal, all 4 angles = 90°, diagonals equal and bisect at 90°.
Rectangle: Opposite sides equal, all 4 angles = 90°, diagonals equal and bisect each other.
Parallelogram: Opposite sides equal and parallel, opposite angles equal, diagonals bisect each other.
Rhombus: All 4 sides equal, opposite angles equal, diagonals bisect each other at 90°.

Minimum data needed:

Square: 1 measurement (side length)
Rectangle: 2 measurements (length and breadth)
Rhombus: 2 measurements (side and one angle, or two diagonals)
Parallelogram: 3 measurements (two sides and included angle, or two sides and a diagonal)

Methods

Method 1: Constructing a Square (given side = a)

Draw a line segment AB = a cm.
At A, construct a 90° angle using compass (perpendicular).
From A, mark point D on this perpendicular such that AD = a cm.
With D as centre and radius = a, draw an arc.
With B as centre and radius = a, draw another arc to intersect the first arc at C.
Join BC and DC. ABCD is the required square.

Method 2: Constructing a Rectangle (given length = l, breadth = b)

Draw AB = l cm (the length).
At A, construct a 90° angle.
From A, mark D on the perpendicular such that AD = b cm (the breadth).
At B, construct a 90° angle.
From B, mark C on this perpendicular such that BC = b cm.
Join DC. ABCD is the required rectangle.

Method 3: Constructing a Parallelogram (given two sides and included angle)

Draw AB = one side.
At A, construct the given included angle.
From A along this ray, mark AD = second side.
With D as centre and radius = AB, draw an arc.
With B as centre and radius = AD, draw an arc intersecting the previous arc at C.
Join BC and DC. ABCD is the required parallelogram.

Method 4: Constructing a Rhombus (given two diagonals d₁ and d₂)

Draw one diagonal AC = d₁ cm.
Find the midpoint O of AC by perpendicular bisector construction.
Through O, draw a line perpendicular to AC.
From O, mark B and D on this perpendicular such that OB = OD = d₂/2.
Join AB, BC, CD, and DA. ABCD is the required rhombus.

Solved Examples

Example 1: Example 1: Construct a square of side 5 cm

Problem: Construct a square ABCD with side 5 cm.

Solution:

Given: Side = 5 cm

Draw AB = 5 cm.
At A, construct ∠DAB = 90° using compass.
With A as centre and radius 5 cm, cut the perpendicular at D. So AD = 5 cm.
With D as centre and radius 5 cm, draw an arc above AB.
With B as centre and radius 5 cm, draw an arc to cut the previous arc at C.
Join BC and DC.

Result: ABCD is the required square with side 5 cm.

Verification: Measure all sides (each = 5 cm) and all angles (each = 90°). Diagonals AC = BD (should be 5√2 ≈ 7.07 cm).

Example 2: Example 2: Construct a rectangle with length 6 cm and breadth 4 cm

Problem: Construct a rectangle PQRS with PQ = 6 cm and QR = 4 cm.

Solution:

Given: Length = 6 cm, Breadth = 4 cm

Draw PQ = 6 cm.
At Q, construct ∠PQR = 90°.
With Q as centre and radius 4 cm, mark R on this perpendicular.
At P, construct ∠QPS = 90°.
With P as centre and radius 4 cm, mark S on this perpendicular.
Join RS.

Result: PQRS is the required rectangle.

Verification: PQ = RS = 6 cm, QR = PS = 4 cm, all angles = 90°.

Example 3: Example 3: Construct a rhombus with diagonals 8 cm and 6 cm

Problem: Construct a rhombus ABCD with diagonals AC = 8 cm and BD = 6 cm.

Solution:

Given: Diagonals: AC = 8 cm, BD = 6 cm

Draw AC = 8 cm.
Find midpoint O of AC using perpendicular bisector construction. So OA = OC = 4 cm.
At O, draw a perpendicular to AC.
On this perpendicular, mark B above AC such that OB = 3 cm (half of BD = 6/2).
On the other side of AC, mark D such that OD = 3 cm.
Join AB, BC, CD, and DA.

Result: ABCD is the required rhombus.

Verification: All four sides should be equal. Side = √(OA² + OB²) = √(4² + 3²) = √(16 + 9) = √25 = 5 cm. So each side = 5 cm.

Example 4: Example 4: Construct a parallelogram with sides 5 cm, 4 cm and included angle 60°

Problem: Construct a parallelogram ABCD with AB = 5 cm, AD = 4 cm, and ∠DAB = 60°.

Solution:

Given: AB = 5 cm, AD = 4 cm, ∠A = 60°

Draw AB = 5 cm.
At A, construct ∠BAX = 60° using compass (standard 60° construction).
On ray AX, mark D such that AD = 4 cm.
With D as centre and radius 5 cm (= AB), draw an arc.
With B as centre and radius 4 cm (= AD), draw an arc to cut the previous arc at C.
Join BC and DC.

Result: ABCD is the required parallelogram.

Verification: AB = DC = 5 cm, AD = BC = 4 cm, ∠A = ∠C = 60°, ∠B = ∠D = 120°.

Example 5: Example 5: Construct a square given diagonal 7 cm

Problem: Construct a square ABCD with diagonal AC = 7 cm.

Solution:

Given: Diagonal = 7 cm

Key property: Diagonals of a square are equal and bisect each other at 90°.

Draw AC = 7 cm.
Find midpoint O of AC by perpendicular bisector construction. OA = OC = 3.5 cm.
At O, draw a perpendicular to AC.
From O, mark B and D on this perpendicular such that OB = OD = 3.5 cm (since diagonals are equal, BD = 7 cm, so OB = 3.5 cm).
Join AB, BC, CD, DA.

Result: ABCD is the required square with diagonal 7 cm.

Verification: Each side = 7/√2 = 7√2/2 ≈ 4.95 cm. All sides should be equal. All angles = 90°.

Example 6: Example 6: Construct a rectangle given one side and diagonal

Problem: Construct a rectangle ABCD with AB = 6 cm and diagonal AC = 10 cm.

Solution:

Given: AB = 6 cm, diagonal AC = 10 cm

Finding breadth: In right triangle ABC, BC² = AC² − AB² = 100 − 36 = 64, so BC = 8 cm.

Draw AB = 6 cm.
At B, construct ∠ABC = 90°.
With B as centre, radius 8 cm, mark C on the perpendicular. BC = 8 cm.
At A, construct ∠DAB = 90°.
With A as centre, radius 8 cm, mark D on this perpendicular. AD = 8 cm.
Join DC.

Result: ABCD is the required rectangle.

Verification: AB = DC = 6 cm, BC = AD = 8 cm. Diagonal AC = √(6² + 8²) = √100 = 10 cm. Correct!

Example 7: Example 7: Construct a parallelogram given two sides and one diagonal

Problem: Construct a parallelogram ABCD with AB = 6 cm, BC = 4 cm, and diagonal AC = 7 cm.

Solution:

Given: AB = 6 cm, BC = 4 cm, AC = 7 cm

Draw AB = 6 cm.
With A as centre and radius 7 cm, draw an arc.
With B as centre and radius 4 cm, draw an arc to cut the first arc at C.
Join AC and BC. Triangle ABC is formed.
With C as centre and radius 6 cm (= AB), draw an arc.
With A as centre and radius 4 cm (= BC), draw an arc to cut the previous arc at D.
Join AD and CD.

Result: ABCD is the required parallelogram.

Verification: AB = CD = 6 cm, BC = AD = 4 cm, AB ∥ DC, AD ∥ BC.

Example 8: Example 8: Construct a rhombus with side 5 cm and one angle 120°

Problem: Construct a rhombus ABCD with side 5 cm and ∠A = 120°.

Solution:

Given: All sides = 5 cm, ∠A = 120°

Draw AB = 5 cm.
At A, construct ∠DAB = 120°.
On this ray, mark D such that AD = 5 cm.
With D as centre and radius 5 cm, draw an arc.
With B as centre and radius 5 cm, draw an arc to cut the previous arc at C.
Join BC and DC.

Result: ABCD is the required rhombus.

Verification: All sides = 5 cm. ∠A = ∠C = 120°, ∠B = ∠D = 60° (opposite angles equal, adjacent supplementary).

Real-World Applications

Real-life applications of constructing special quadrilaterals:

Architecture and floor plans: Rooms, windows, and doors are typically rectangular. Accurate construction ensures proper fit.
Tile and flooring design: Tiles come in square, rectangular, and rhombus shapes. Laying patterns require precise construction.
Engineering drawings: Machine parts, circuit boards, and mechanical components often use parallelogram and rhombus shapes.
Land surveying: Plotting rectangular and square land parcels requires accurate construction with measured dimensions.
Art and design: Geometric patterns in rangoli, quilting, and graphic design use special quadrilaterals.
Carpentry: Ensuring rectangular frames are truly rectangular (checking diagonals are equal) is a practical construction skill.

Key Points to Remember

A square can be constructed with just one measurement (side or diagonal).
A rectangle needs two measurements (length and breadth, or one side and diagonal).
A rhombus needs two measurements (side and one angle, or two diagonals).
A parallelogram needs three measurements (two sides and included angle, or two sides and a diagonal).
Always start by drawing the base line segment.
Use the compass for equal lengths and perpendicular bisector constructions.
For 90° angles: construct a perpendicular using the compass arc method.
For 60° angles: use the standard equilateral triangle construction.
Diagonals of a rhombus bisect each other at right angles — this is key for the diagonal method.
Always verify the construction by measuring sides, angles, and diagonals.

Practice Problems

Construct a square ABCD with side 4.5 cm.
Construct a rectangle PQRS with PQ = 7 cm and QR = 3 cm.
Construct a rhombus with diagonals 10 cm and 8 cm. Also find the side length.
Construct a parallelogram with sides 6 cm and 4.5 cm and included angle 75°.
Construct a square with diagonal 6 cm. Measure and verify the side length.
Construct a rectangle with one side 5 cm and diagonal 13 cm. Find the other side.
Construct a rhombus ABCD with side 6 cm and ∠A = 60°. Measure the diagonals.

Frequently Asked Questions

Q1. How many measurements are needed to construct a square?

Only one measurement is needed — either the side length or the diagonal length. Since all sides are equal and all angles are 90°, one measurement completely determines the square.

Q2. Why do diagonals of a rhombus bisect at right angles?

In a rhombus, all four sides are equal. This makes the diagonals split the rhombus into four congruent right triangles. Because of this symmetry, the diagonals must cross at 90° and each diagonal is bisected by the other.

Q3. Can a parallelogram be constructed with just two sides?

No. Two sides are not enough because the angle between them can vary, producing different parallelograms. You need a third measurement — either the included angle or a diagonal — to fix the shape uniquely.

Q4. How do I construct a 90° angle using a compass?

Draw a line and mark a point. With the point as centre, draw an arc cutting the line at two points. From each of these points, draw arcs of equal radius above the line. The intersection of these arcs, joined to the original point, gives a 90° angle.

Q5. What is the difference between constructing a rhombus using sides and using diagonals?

When given a side and an angle, you construct the rhombus like a parallelogram (all sides equal). When given two diagonals, you use the property that diagonals bisect each other at 90° — draw one diagonal, bisect it, draw the perpendicular, and mark the other diagonal's halves.

Q6. How do I verify my construction is correct?

Measure all sides and angles using a ruler and protractor. For a square, check all sides are equal and all angles are 90°. For a rectangle, check opposite sides are equal and all angles are 90°. For a rhombus, check all sides are equal. Also verify that diagonals have the expected properties.

Q7. Is a square a special type of rhombus?

Yes. A square has all four sides equal (like a rhombus) AND all angles are 90°. So a square is a rhombus with the additional property of right angles. Similarly, a square is also a special rectangle.

Q8. Can I construct a parallelogram using two diagonals?

Not uniquely. Knowing only the two diagonals of a parallelogram does not fix its shape because the angle between the diagonals can vary. You would also need to know the angle between the diagonals or some side length.

Constructing Special Quadrilaterals

What is Constructing Special Quadrilaterals?

Methods

Solved Examples

Example 1: Example 1: Construct a square of side 5 cm

Example 2: Example 2: Construct a rectangle with length 6 cm and breadth 4 cm

Example 3: Example 3: Construct a rhombus with diagonals 8 cm and 6 cm

Example 4: Example 4: Construct a parallelogram with sides 5 cm, 4 cm and included angle 60°

Example 5: Example 5: Construct a square given diagonal 7 cm

Example 6: Example 6: Construct a rectangle given one side and diagonal

Example 7: Example 7: Construct a parallelogram given two sides and one diagonal

Example 8: Example 8: Construct a rhombus with side 5 cm and one angle 120°

Real-World Applications

Key Points to Remember

Practice Problems

Frequently Asked Questions

Q1. How many measurements are needed to construct a square?

Q2. Why do diagonals of a rhombus bisect at right angles?

Q3. Can a parallelogram be constructed with just two sides?

Q4. How do I construct a 90° angle using a compass?

Q5. What is the difference between constructing a rhombus using sides and using diagonals?

Q6. How do I verify my construction is correct?

Q7. Is a square a special type of rhombus?

Q8. Can I construct a parallelogram using two diagonals?

Related Topics

More Maths Topics

Quick Links