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Constructing Quadrilaterals

Class 8Practical Geometry

In Class 8 Practical Geometry, you learn how to construct quadrilaterals using a ruler, compass, and protractor. To uniquely construct a quadrilateral, you need a specific number of independent measurements. A quadrilateral has 4 sides, 4 angles, and 2 diagonals — a total of 10 elements. However, you do not need all 10 to draw a unique quadrilateral.

A triangle requires 3 independent measurements (SSS, SAS, ASA). A quadrilateral, being a more complex shape, requires 5 independent measurements to construct it uniquely. These 5 measurements can come in different combinations — 4 sides and 1 diagonal, 3 sides and 2 diagonals, and so on.

This topic is important because it connects geometric theory with hands-on construction skills. Accurate construction requires careful use of compass arcs, perpendicular bisectors, and angle measurements.

What is Constructing Quadrilaterals?

Definition: Construction of a quadrilateral means drawing a quadrilateral of exact dimensions using a ruler, compass, and protractor, given sufficient measurements.


Minimum measurements needed: 5 independent measurements to uniquely determine a quadrilateral.


Common combinations of 5 measurements:

  • Case 1: 4 sides and 1 diagonal
  • Case 2: 3 sides and 2 diagonals
  • Case 3: 4 sides and 1 angle
  • Case 4: 3 sides and 2 included angles
  • Case 5: 2 sides and 3 angles

Principle: In each case, the strategy is to divide the quadrilateral into two triangles using a diagonal, construct one triangle first, then complete the other triangle to form the quadrilateral.

Constructing Quadrilaterals Formula

No formula is used for construction — instead, step-by-step geometric procedures are followed.


Tools required:

  • Ruler — for drawing straight lines of required length
  • Compass — for drawing arcs and transferring lengths
  • Protractor — for measuring and drawing angles
  • Pencil — sharp, for accuracy

Key geometric facts used in constructions:

  • Two arcs from different centres intersect at a unique point (if a solution exists).
  • A triangle can be constructed if three independent measurements are known (SSS, SAS, or ASA).
  • The diagonal divides a quadrilateral into two triangles.
  • Angle sum of a quadrilateral = 360 degrees (useful when only some angles are given).

Derivation and Proof

Why are 5 measurements needed?

A quadrilateral ABCD has 4 vertices, each with 2 coordinates (x, y). That gives 8 unknowns. But a quadrilateral can be placed anywhere on the plane, and its position and orientation do not affect its shape. Fixing one vertex removes 2 unknowns, and fixing the direction of one side removes 1 more. So the number of unknowns that determine the shape = 8 - 3 = 5.


Why the diagonal method works:

A diagonal divides the quadrilateral into 2 triangles. Each triangle needs 3 measurements to construct. But the diagonal is shared between both triangles, so it counts for both. If we know 3 measurements for the first triangle (say, two sides and the diagonal), the diagonal is now fixed. For the second triangle, we need 2 more measurements (the remaining two sides, or one side and one angle). Total = 3 + 2 = 5 measurements.

Types and Properties

The five standard cases for constructing a quadrilateral are:

Case 1: 4 sides and 1 diagonal

  1. Draw the diagonal.
  2. Construct one triangle using the diagonal and two sides.
  3. Construct the other triangle using the diagonal and the remaining two sides.

Case 2: 3 sides and 2 diagonals

  1. Draw one side.
  2. Use the two diagonals and remaining sides to locate the other vertices using compass arcs.

Case 3: 4 sides and 1 angle

  1. Draw one side.
  2. Construct the given angle at the appropriate vertex.
  3. Mark the adjacent side along the angle ray.
  4. Use compass arcs with the remaining side lengths to find the last vertex.

Case 4: 3 sides and 2 included angles

  1. Draw one side.
  2. Construct the two given angles at the endpoints of that side.
  3. Mark the other sides along the angle rays.
  4. Join the final vertices.

Case 5: 2 sides and 3 angles

  1. Calculate the 4th angle using the angle sum property (360 degrees).
  2. Draw one side.
  3. Construct the given angles at appropriate vertices.
  4. The intersection of the rays gives the remaining vertices.

Solved Examples

Example 1: Example 1: Construct quadrilateral ABCD — 4 sides and 1 diagonal

Problem: Construct quadrilateral ABCD where AB = 4 cm, BC = 5 cm, CD = 6 cm, DA = 5.5 cm, and diagonal AC = 7 cm.


Steps:

  1. Draw AC = 7 cm.
  2. With A as centre and radius 4 cm, draw an arc.
  3. With C as centre and radius 5 cm, draw an arc. The intersection of these arcs is B.
  4. Join AB and BC.
  5. With A as centre and radius 5.5 cm, draw an arc on the other side of AC.
  6. With C as centre and radius 6 cm, draw an arc. The intersection is D.
  7. Join AD and CD.

ABCD is the required quadrilateral.

Example 2: Example 2: Construct quadrilateral — 3 sides and 2 diagonals

Problem: Construct quadrilateral PQRS where PQ = 4 cm, QR = 6 cm, PR = 7 cm, QS = 5.5 cm, and RS = 5 cm.


Steps:

  1. Draw PQ = 4 cm.
  2. With P as centre and radius 7 cm, and Q as centre and radius 6 cm, draw arcs to get R.
  3. Join PR and QR. Triangle PQR is complete.
  4. With Q as centre and radius 5.5 cm, and R as centre and radius 5 cm, draw arcs to get S (on the other side of QR from P).
  5. Join PS, QS, and RS.

PQRS is the required quadrilateral.

Example 3: Example 3: Construct quadrilateral — 4 sides and 1 angle

Problem: Construct quadrilateral ABCD where AB = 5 cm, BC = 4 cm, CD = 6 cm, DA = 5 cm, and angle A = 100 degrees.


Steps:

  1. Draw AB = 5 cm.
  2. At A, construct an angle of 100 degrees using a protractor.
  3. Along the angle ray from A, mark D such that AD = 5 cm.
  4. With B as centre and radius 4 cm, draw an arc.
  5. With D as centre and radius 6 cm, draw an arc. The intersection is C.
  6. Join BC and DC.

ABCD is the required quadrilateral.

Example 4: Example 4: Construct quadrilateral — 3 sides and 2 angles

Problem: Construct quadrilateral ABCD where AB = 5 cm, BC = 4.5 cm, CD = 6 cm, angle B = 105 degrees, and angle C = 80 degrees.


Steps:

  1. Draw BC = 4.5 cm.
  2. At B, construct an angle of 105 degrees. Along this ray, mark A such that BA = 5 cm.
  3. At C, construct an angle of 80 degrees on the same side. Along this ray, mark D such that CD = 6 cm.
  4. Join AD.

ABCD is the required quadrilateral.

Example 5: Example 5: Construct quadrilateral — 2 sides and 3 angles

Problem: Construct quadrilateral PQRS where PQ = 5 cm, QR = 4 cm, angle P = 80 degrees, angle Q = 110 degrees, and angle R = 85 degrees.


Steps:

  1. Calculate angle S = 360 - (80 + 110 + 85) = 85 degrees.
  2. Draw PQ = 5 cm.
  3. At Q, construct angle Q = 110 degrees. Along this ray, mark R such that QR = 4 cm.
  4. At P, construct angle P = 80 degrees.
  5. At R, construct angle R = 85 degrees.
  6. The rays from P and R intersect at S.

PQRS is the required quadrilateral.

Example 6: Example 6: Construct a rectangle

Problem: Construct a rectangle ABCD with length 6 cm and breadth 4 cm.


Steps:

  1. Draw AB = 6 cm.
  2. At A, construct a 90-degree angle. Mark D on this ray such that AD = 4 cm.
  3. At B, construct a 90-degree angle. Mark C on this ray such that BC = 4 cm.
  4. Join DC. (DC should equal 6 cm; verify.)

ABCD is the required rectangle. This is Case 4 (3 sides + 2 right angles).

Example 7: Example 7: Construct a rhombus

Problem: Construct a rhombus ABCD with side 5 cm and diagonal AC = 8 cm.


Steps:

  1. Draw AC = 8 cm.
  2. With A as centre and radius 5 cm, draw an arc above AC.
  3. With C as centre and radius 5 cm, draw an arc above AC. The intersection is B.
  4. Similarly, draw arcs below AC with the same radii. The intersection is D.
  5. Join AB, BC, CD, and DA.

ABCD is the required rhombus. (This uses 4 equal sides + 1 diagonal.)

Example 8: Example 8: Construct a square

Problem: Construct a square ABCD with side 4.5 cm.


Steps:

  1. Draw AB = 4.5 cm.
  2. At A, construct a 90-degree angle. Mark D on this ray such that AD = 4.5 cm.
  3. With B as centre and radius 4.5 cm, and D as centre and radius 4.5 cm, draw arcs. The intersection is C.
  4. Join BC and DC.

ABCD is the required square.

Example 9: Example 9: Construct a parallelogram

Problem: Construct a parallelogram ABCD where AB = 6 cm, BC = 4 cm, and angle B = 75 degrees.


Steps:

  1. Draw AB = 6 cm.
  2. At B, construct an angle of 75 degrees. Mark C on this ray such that BC = 4 cm.
  3. With A as centre and radius 4 cm (= BC), draw an arc.
  4. With C as centre and radius 6 cm (= AB), draw an arc. The intersection is D.
  5. Join AD and CD.

ABCD is the required parallelogram. (Opposite sides of a parallelogram are equal, so we need only 2 sides and 1 angle — 3 measurements, but quadrilateral needs 5. The parallelogram constraints provide the other 2.)

Example 10: Example 10: Construction not possible

Problem: Can you construct quadrilateral ABCD with AB = 3 cm, BC = 4 cm, CD = 5 cm, DA = 6 cm, and diagonal AC = 10 cm? Explain.


Solution:

For triangle ABC: AB + BC = 3 + 4 = 7 cm. But AC = 10 cm.

By the triangle inequality, the sum of any two sides must be greater than the third side. Here, 7 < 10, which violates this condition.

Therefore, triangle ABC cannot be constructed, and hence the quadrilateral cannot be constructed.

Answer: Construction is not possible because the given measurements violate the triangle inequality.

Real-World Applications

Constructing quadrilaterals has many practical uses:

  • Architecture: Architects draw floor plans, room layouts, and building footprints that are quadrilateral shapes. Accurate construction ensures correct dimensions.
  • Engineering: Machine parts, brackets, and frames often have quadrilateral shapes that must be drawn to precise measurements.
  • Surveying: Land plots are often quadrilateral. Surveyors use measurements of sides and diagonals to draw accurate plot maps.
  • Carpentry: Wooden frames, doors, and windows require precise quadrilateral shapes. Carpenters use the principles of geometric construction.
  • Art and Design: Geometric patterns in art, tiling, and fabric design use accurately constructed quadrilaterals.
  • Computer-Aided Design (CAD): Modern software automates construction, but the underlying geometric principles remain the same as manual construction.

Key Points to Remember

  • 5 independent measurements are needed to construct a unique quadrilateral.
  • The key strategy is to divide the quadrilateral into two triangles using a diagonal.
  • Common cases: 4 sides + 1 diagonal, 3 sides + 2 diagonals, 4 sides + 1 angle, 3 sides + 2 angles, 2 sides + 3 angles.
  • Always draw a rough sketch first to plan the construction.
  • Use the triangle inequality to check if the construction is possible.
  • A rectangle needs only length and breadth (the 90-degree angles provide the other constraints).
  • A parallelogram needs only 2 sides and 1 angle (parallel and equal sides provide the rest).
  • A square needs only the side length (equal sides and right angles fix everything).
  • If 3 angles are given, the 4th can be found using the angle sum property (360 degrees).
  • Compass arcs must be drawn with the correct radius from the correct centre.

Practice Problems

  1. Construct quadrilateral ABCD where AB = 4.5 cm, BC = 5 cm, CD = 5.5 cm, DA = 4 cm, and diagonal AC = 6.5 cm.
  2. Construct quadrilateral PQRS where PQ = 5 cm, QR = 4 cm, RS = 6 cm, angle Q = 100 degrees, and angle R = 90 degrees.
  3. Construct a parallelogram with sides 5 cm and 3.5 cm, and one angle 60 degrees.
  4. Construct a rhombus with side 4 cm and one diagonal 5 cm.
  5. Construct a rectangle with length 7 cm and breadth 3 cm.
  6. Construct quadrilateral ABCD where AB = 5 cm, BC = 6 cm, AC = 7 cm, AD = 4 cm, and BD = 6.5 cm.
  7. Construct a quadrilateral where PQ = 4 cm, QR = 5 cm, angle P = 70 degrees, angle Q = 100 degrees, and angle R = 80 degrees.
  8. Can a quadrilateral be constructed with sides 2, 3, 4, 5 cm and diagonal 12 cm? Explain.

Frequently Asked Questions

Q1. How many measurements are needed to construct a unique quadrilateral?

5 independent measurements are needed. These can be combinations of sides, diagonals, and angles.

Q2. Why do we need 5 measurements for a quadrilateral but only 3 for a triangle?

A triangle has 3 sides and 3 angles (6 elements), but 3 measurements fix it due to the angle sum property. A quadrilateral has more elements and more degrees of freedom, requiring 5 measurements.

Q3. What is the basic strategy for constructing a quadrilateral?

Divide the quadrilateral into two triangles using a diagonal. Construct one triangle first, then use the remaining measurements to complete the second triangle.

Q4. What tools are needed for construction?

A ruler (for drawing lines), a compass (for arcs and circles), a protractor (for measuring angles), and a sharp pencil.

Q5. Can every set of 5 measurements produce a valid quadrilateral?

No. The measurements must satisfy the triangle inequality for each triangle formed by the diagonal. If they do not, construction is not possible.

Q6. How do you construct a square if only the side is given?

A square has equal sides and all 90-degree angles. So knowing just the side (1 measurement) plus the constraints (equal sides, right angles) gives 5 effective measurements. Draw one side, construct 90-degree angles at both ends, mark equal sides, and join.

Q7. Why should we draw a rough sketch first?

A rough sketch helps plan the order of steps, identify which triangle to construct first, and check whether the measurements are reasonable before starting the actual construction.

Q8. What if 3 angles are given and we need the 4th?

Use the angle sum property: the sum of all angles of a quadrilateral is 360 degrees. The 4th angle = 360 - (sum of the 3 given angles).

Related Topics

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