MathsICSE Class 8 · Chapter 17

Construction of Quadrilaterals ICSE Class 8: The Four Cases, Step by Step

Q: Why can't four sides alone fix a quadrilateral?

Because the shape can change while the sides stay the same. A square can be pushed into a rhombus without changing any side.

Q: Can I use a protractor for all angles?

You may use a protractor when the angle cannot be made with compasses, but where an angle such as 60° or 90° can be constructed, the exam may expect compasses.

Q: What if my measured answer is slightly different from the expected one?

Small differences (about 1 mm or 1°) come from drawing. Larger differences mean an error, so check your arcs and angles.

Q: Which triangle should I draw first?

One for which all three needed parts are given. In Case 1 that is either triangle on the diagonal; in Case 3 it is the side between the two angles.

Q: Do I need to write the steps of construction?

Yes, usually. Number the steps clearly, in the order you carried them out.

Updated 15 October 2026 · Fact-checked

A quadrilateral can be constructed uniquely from five suitable measurements. ICSE Class 8 covers four cases: four sides and one diagonal; three sides and two diagonals; three sides and the two included angles; and two adjacent sides with three angles. Each starts with a rough sketch and builds one triangle first.

What Construction of Quadrilaterals is about

To construct a figure means to draw it accurately using a ruler, compasses and a protractor, from given measurements.

A triangle is fixed by 3 measurements. A quadrilateral needs 5: four equal sides alone do not fix a shape, because a square and a rhombus can have the same sides. The trick in every case is to split the quadrilateral into triangles using a diagonal, construct one triangle, and then locate the fourth vertex.

The CISCE Class VIII curriculum asks you to construct a quadrilateral given:
1. four sides and one diagonal;
2. three sides and two diagonals;
3. three sides and two included angles;
4. two adjacent sides and three angles.

In a school exam you write the steps of construction, draw neatly, keep all construction arcs, and often measure another side, diagonal or angle at the end. Start every construction with a rough sketch with the measurements marked on it.

Weightage: CISCE publishes no question paper, marking scheme or chapter weightage for Class VIII Mathematics: your school sets and marks the exam. Construction of Quadrilaterals sits in the Geometry theme of the CISCE Curriculum for Upper Primary Classes (VI-VIII).Source: CISCE Curriculum for Upper Primary Classes (VI-VIII), RDCD 2016

Key concepts

Why five measurements, and the rough sketch

Four sides of 4 cm can make a square or many different rhombuses: the shape "wobbles". Fixing one more measurement (a diagonal or an angle) stops the wobble.

Before constructing:
1. Draw a rough sketch and write the given measurements on it.
2. Decide which triangle can be drawn first (one with three known parts).
3. Check that each triangle can exist: the sum of any two sides must be greater than the third side.

Always write the steps in order and leave all arcs visible.

Worked example

Can a quadrilateral $A B C D$ be drawn with $A B = 4$ cm, $B C = 3$ cm and diagonal $A C = 8$ cm?

Try it first, then show the solutionSolution

Triangle $A B C$ must be drawn first, with sides 4, 3 and 8 cm.
$4 + 3 = 7 < 8$ , so the arcs from $A$ and $C$ will not meet.
The triangle, and so the quadrilateral, cannot be drawn.

Answer: No: $4 + 3$ is less than 8.

Case 1: four sides and one diagonal

The diagonal splits $A B C D$ into two triangles whose three sides are all known.

Steps (given $A B$ , $B C$ , $C D$ , $D A$ and $A C$ ):
1. Draw $A C$ .
2. With centre $A$ , radius $A B$ , draw an arc; with centre $C$ , radius $C B$ , draw an arc cutting it at $B$ (on one side of $A C$ ).
3. With centre $A$ , radius $A D$ , and centre $C$ , radius $C D$ , draw arcs meeting at $D$ on the other side of $A C$ .
4. Join $A B$ , $B C$ , $C D$ , $D A$ .

Worked example

Construct $A B C D$ with $A B = 4$ cm, $B C = 6$ cm, $C D = 5$ cm, $D A = 5.5$ cm and $A C = 7$ cm. Measure $B D$ .

Try it first, then show the solutionSolution

Draw $A C = 7$ cm.
Arcs of 4 cm from $A$ and 6 cm from $C$ meet at $B$ below $A C$ .
Arcs of 5.5 cm from $A$ and 5 cm from $C$ meet at $D$ above $A C$ ; join the sides.
Measure $B D$ : about 7.5 cm.

Answer: $B D \approx 7.5$ cm

Case 2: three sides and two diagonals

Given $A B$ , $B C$ , $C D$ and the diagonals $A C$ and $B D$ :
1. Construct triangle $A B C$ from $A B$ , $B C$ and $A C$ .
2. With centre $C$ , radius $C D$ , draw an arc; with centre $B$ , radius $B D$ , draw an arc cutting it at $D$ . Choose the intersection on the opposite side of $A C$ from $B$ .
3. Join $C D$ and $D A$ .

The fourth side $D A$ is not given; it comes out of the construction.

Worked example

Construct $A B C D$ with $A B = 5$ cm, $B C = 4.5$ cm, $C D = 6$ cm, $A C = 7$ cm and $B D = 6.5$ cm. Measure $D A$ .

Try it first, then show the solutionSolution

Construct triangle $A B C$ : $A C = 7$ cm, arcs of 5 cm from $A$ and 4.5 cm from $C$ .
From $C$ an arc of 6 cm; from $B$ an arc of 6.5 cm; they meet at $D$ across $A C$ from $B$ .
Join $C D$ and $D A$ ; $D A$ measures about 3.5 cm.

Answer: $D A \approx 3.5$ cm

Case 3: three sides and two included angles

Given $A B$ , $B C$ , $C D$ and the angles $∠ B$ and $∠ C$ (the angles between these sides):
1. Draw the middle side $B C$ .
2. At $B$ , construct $∠ B$ and cut off $B A$ on that ray.
3. At $C$ , construct $∠ C$ and cut off $C D$ on that ray (both rays on the same side of $B C$ ).
4. Join $A D$ .

Use compasses for $60°$ , $90°$ , $120°$ , $45°$ , $30°$ , $75°$ , $105°$ where possible; otherwise a protractor.

Worked example

Construct $A B C D$ with $A B = 5$ cm, $B C = 4$ cm, $C D = 4.5$ cm, $∠ B = 75°$ and $∠ C = 120°$ . Measure $A D$ .

Try it first, then show the solutionSolution

Draw $B C = 4$ cm.
At $B$ make $75°$ and mark $A$ with $B A = 5$ cm.
At $C$ make $120°$ on the same side and mark $D$ with $C D = 4.5$ cm.
Join $A D$ : about 5 cm.

Answer: $A D \approx 5$ cm

Case 4: two adjacent sides and three angles

Given $A B$ , $B C$ and $∠ A$ , $∠ B$ , $∠ C$ :
1. Find $∠ D = 360° - (∠ A + ∠ B + ∠ C)$ as a check (it must be positive).
2. Draw $A B$ ; at $B$ construct $∠ B$ and cut off $B C$ .
3. At $A$ construct $∠ A$ ; at $C$ construct $∠ C$ . The two rays meet at $D$ .
4. Measure $∠ D$ to check your drawing.

Worked example

Construct $A B C D$ with $A B = 5.5$ cm, $B C = 4$ cm, $∠ A = 60°$ , $∠ B = 105°$ and $∠ C = 105°$ .

Try it first, then show the solutionSolution

$∠ D = 360° - 270° = 90°$ (a useful check).
Draw $A B = 5.5$ cm; at $B$ make $105°$ and mark $C$ with $B C = 4$ cm.
At $A$ make $60°$ ; at $C$ make $105°$ ; the rays meet at $D$ .
Check: $∠ D$ measures $90°$ ; $A D \approx 6.6$ cm and $C D \approx 3.7$ cm.

Answer: $∠ D = 90°$ ; $A D \approx 6.6$ cm, $C D \approx 3.7$ cm

Special quadrilaterals

Use the properties from Understanding Quadrilaterals to make fewer measurements enough:

Parallelogram: two adjacent sides and the angle between them (opposite sides are equal, so draw arcs for the fourth vertex).
Rectangle: two adjacent sides (angles are $90°$ ).
Square: one side, or one diagonal (diagonals are equal and bisect at right angles).
Rhombus: two diagonals (draw one, then its perpendicular bisector, and mark half the other diagonal on each side).

Worked example

Construct a rhombus with diagonals 6 cm and 8 cm, and measure its side.

Try it first, then show the solutionSolution

Draw $A C = 8$ cm and construct its perpendicular bisector, meeting $A C$ at $O$ .
Mark $B$ and $D$ on the bisector, 3 cm from $O$ on each side.
Join $A B$ , $B C$ , $C D$ , $D A$ ; each side measures 5 cm.

Answer: Each side 5 cm

Maths in real life: Construction of Quadrilaterals

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Checking a rectangle

Suppose you build a rectangular picture frame 30 cm by 20 cm and want to check the corners are square.

What should each diagonal measure?

Try it first, then show the solutionSolution

In a rectangle, both diagonals are equal.
A scale drawing (or calculation) gives a diagonal of about 36.1 cm; if both diagonals match this, the corners are right angles.

Answer: About 36.1 cm each

Memory hook: Equal diagonals mean a true rectangle.

Splitting into triangles

Imagine a surveyor measures a four-sided field: sides 20 m, 15 m, 18 m and 16 m, and one diagonal 25 m between the corners joining the 20 m and 15 m sides.

Can each triangle be drawn?

Try it first, then show the solutionSolution

Triangle 1: 20, 15, 25: $20 + 15 = 35 > 25$ ✓.
Triangle 2: 18, 16, 25: $18 + 16 = 34 > 25$ ✓. Both can be drawn, so the field can be mapped to scale.

Answer: Yes, both triangles exist.

Memory hook: A diagonal turns a quadrilateral into two triangles.

Why four sides are not enough

Suppose four sticks of 10 cm are hinged at the corners to make a frame.

Why does the frame wobble, and what fixes it as a square?

Try it first, then show the solutionSolution

Equal sides allow a square or any rhombus, so the shape is not fixed.
Fixing a diagonal at about 14.1 cm (the diagonal of a 10 cm square) locks it as a square.

Answer: A fifth measurement, such as a diagonal, fixes it.

Memory hook: Five measurements fix a quadrilateral.

Topics in this chapter

Constructing Quadrilaterals from Five Measurements

Construction of Quadrilaterals formula sheet

Measurements needed: $5 independent measurements$; Fewer for special shapes.
Triangle check: $a + b > c$; For every triangle you construct.
Fourth angle: $∠ D = 360° - (∠ A + ∠ B + ∠ C)$; Case 4 check.

How the school exam asks Construction of Quadrilaterals

The question patterns to expect, and the method that works for each.

Question type	How it is asked	Method
Four sides and a diagonal	Construct ABCD with given sides and AC; measure BD.	Diagonal first, then arcs for B and D on opposite sides.
Three sides and two diagonals	Construct ABCD given AB, BC, CD, AC and BD.	Triangle ABC, then D from arcs about C and B.
Three sides and two included angles	Construct ABCD given AB, BC, CD, ∠B and ∠C.	Middle side first; angles at both ends; join the free ends.
Two sides and three angles	Construct ABCD given AB, BC, ∠A, ∠B, ∠C.	Check ∠D; draw AB and BC; angle rays from A and C meet at D.
Special shapes and feasibility	Construct a rhombus from its diagonals; say whether a quadrilateral can be drawn.	Use the shape's properties; test each triangle.

Tricks to remember Construction of Quadrilaterals

Short memory aids. Say each one aloud once, then use it on a question the same day.

Sketch, triangle, fourth point
Every construction: rough sketch, build one triangle, then find the fourth vertex.
Use it when: Planning any case.
Five fixes the shape
Four measurements wobble; the fifth one locks the quadrilateral.
Use it when: Knowing why each case gives 5 measurements.
Opposite sides of the diagonal
In Case 1, B and D go on opposite sides of AC, or the figure crosses itself.
Use it when: Placing arcs correctly.
Arcs stay
Never rub out construction arcs: they are your working.
Use it when: Getting the marks for method.

Construction of Quadrilaterals: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Example 1: How many measurements · 1 mark · MCQExam level

The number of independent measurements needed to construct a general quadrilateral is:
(a) 3
(b) 4
(c) 5
(d) 6

Try it first, then show the solutionSolution

A triangle needs 3; four sides alone allow a wobbling shape.
One more measurement fixes it: 5.

Answer: (c)

Example 2: Case 1 construction · 3 marks · Short answerExam level

Write the steps to construct $A B C D$ with $A B = 4$ cm, $B C = 6$ cm, $C D = 5$ cm, $D A = 5.5$ cm and $A C = 7$ cm, and state what $B D$ should measure.

Try it first, then show the solutionSolution

Draw $A C = 7$ cm. Arcs of radius 4 cm (centre $A$ ) and 6 cm (centre $C$ ) meet at $B$ .
On the other side of $A C$ , arcs of 5.5 cm (centre $A$ ) and 5 cm (centre $C$ ) meet at $D$ .
Join $A B$ , $B C$ , $C D$ , $D A$ . $B D \approx 7.5$ cm.

Answer: Constructed; $B D \approx 7.5$ cm

Example 3: The missing angle first · 3 marks · Short answerExam level

To construct $A B C D$ with $∠ A = 60°$ , $∠ B = 105°$ and $∠ C = 105°$ , what is $∠ D$ , and why is it worth finding first?

Try it first, then show the solutionSolution

$∠ D = 360° - (60° + 105° + 105°)$ .
$= 90°$ .
It confirms the data is possible (positive angle) and lets you check the drawing by measuring $∠ D$ .

Answer: $90°$

Example 4: Feasibility · 3 marks · Short answerHarder exam level

Can quadrilateral $A B C D$ be constructed with $A B = 4$ cm, $B C = 3$ cm, $C D = 5$ cm, $D A = 6$ cm and $A C = 8$ cm? Explain.

Try it first, then show the solutionSolution

Triangle $A B C$ has sides 4, 3 and 8 cm.
$4 + 3 = 7$ , which is less than 8.
Triangle $A B C$ cannot be drawn, so neither can $A B C D$ .

Answer: No

Example 5: Case 3 construction · 4 marks · Long answerHarder exam level

Construct $A B C D$ with $A B = 5$ cm, $B C = 4$ cm, $C D = 4.5$ cm, $∠ B = 75°$ and $∠ C = 120°$ . Write the steps and measure $A D$ and $∠ A$ .

Try it first, then show the solutionSolution

Draw $B C = 4$ cm.
At $B$ construct $75°$ (bisect between $60°$ and $90°$ ) and cut off $B A = 5$ cm.
At $C$ construct $120°$ on the same side and cut off $C D = 4.5$ cm.
Join $A D$ . Measurements: $A D \approx 5$ cm, $∠ A \approx 94°$ .

Answer: $A D \approx 5$ cm; $∠ A \approx 94°$

Example 6: Case 2 construction · 4 marks · Long answerHarder exam level

Construct $A B C D$ with $A B = 5$ cm, $B C = 4.5$ cm, $C D = 6$ cm, $A C = 7$ cm and $B D = 6.5$ cm. Write the steps and measure $A D$ .

Try it first, then show the solutionSolution

Rough sketch with all five measurements.
Construct triangle $A B C$ : $A C = 7$ cm; arcs 5 cm from $A$ and 4.5 cm from $C$ give $B$ .
Arc 6 cm from $C$ and arc 6.5 cm from $B$ meet at $D$ , on the side of $A C$ away from $B$ .
Join $C D$ and $D A$ ; $A D \approx 3.5$ cm.

Answer: $A D \approx 3.5$ cm

Example 7: A rhombus from its diagonals · 5 marks · Long answerHOTS / competency

Construct a rhombus $A B C D$ with diagonals $A C = 8$ cm and $B D = 6$ cm. Explain why your method works and measure a side.

Try it first, then show the solutionSolution

The diagonals of a rhombus bisect each other at right angles.
Draw $A C = 8$ cm and construct its perpendicular bisector, meeting $A C$ at $O$ (its mid-point).
On the bisector, mark $B$ and $D$ with $O B = O D = 3$ cm.
Join $A B$ , $B C$ , $C D$ , $D A$ .
The diagonals bisect each other (so it is a parallelogram) and are perpendicular (so it is a rhombus). Each side measures 5 cm.

Answer: Side 5 cm

Case-study practice (Section B style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 10-mark question split into 3 + 3 + 4 marks.

Planning a flower bed

Suppose a gardener plans a four-sided flower bed $P QR S$ with $P Q = 5$ m, $QR = 4$ m, $R S = 6$ m, $S P = 4.5$ m and diagonal $P R = 7$ m. She makes a scale drawing using 1 cm for 1 m.

(i) · 3 marks
Which construction case is this, and which line should be drawn first? Why?
Answer
Answer: Four sides and one diagonal; draw PR first
Four sides and a diagonal are given. $P R$ is common to triangles $P QR$ and $P R S$ , so draw $P R = 7$ cm first.
(ii) · 3 marks
Check that both triangles can be drawn.
Answer
Answer: Both can be drawn
Triangle $P QR$ : $5 + 4 = 9 > 7$ ✓. Triangle $P R S$ : $6 + 4.5 = 10.5 > 7$ ✓ (other pairs also exceed the third side).
(iii) · 4 marks
After constructing, she measures $QS$ on the drawing as about 6.9 cm. What is the real distance, and how could she use it?
Answer
Answer: About 6.9 m
Scale 1 cm : 1 m, so about 6.9 m. Measuring this second diagonal on the ground checks that the bed has been marked out the same shape as the drawing.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Write the steps to construct $A B C D$ with $A B = 5.5$ cm, $B C = 4$ cm, $∠ A = 60°$ , $∠ B = 105°$ , $∠ C = 105°$ . (3 marks)

Step	Marks
Find $∠ D = 90°$ and draw a rough sketch	1
Draw $A B = 5.5$ cm; construct $105°$ at $B$ and cut off $B C = 4$ cm	1
Construct $60°$ at $A$ and $105°$ at $C$ ; the rays meet at $D$	1

Construct $A B C D$ with $A B = 4$ cm, $B C = 6$ cm, $C D = 5$ cm, $D A = 5.5$ cm and $A C = 7$ cm. (4 marks)

Step	Marks
Rough sketch and $A C = 7$ cm drawn	1
$B$ located by arcs of 4 cm and 6 cm	1
$D$ located by arcs of 5.5 cm and 5 cm on the other side of $A C$	1
Sides joined, arcs visible, vertices labelled	1

Construct $A B C D$ with $A B = 4.5$ cm, $B C = 5.5$ cm, $C D = 4$ cm, $∠ B = 60°$ and $∠ C = 90°$ . Measure $A D$ . (5 marks)

Step	Marks
Rough sketch with the given parts	1
$B C = 5.5$ cm drawn	1
$60°$ constructed at $B$ with compasses; $B A = 4.5$ cm cut off	1
$90°$ constructed at $C$ ; $C D = 4$ cm cut off	1
$A D$ joined and measured: about 3.3 cm	1

Where students lose marks in Construction of Quadrilaterals

The slips examiners see most often, and what to write instead.

Mistake: No rough sketch
Without a sketch it is easy to put a vertex on the wrong side or use the wrong angle.
Do this: Always draw and label a rough figure first.
Usually costs: Often the whole construction.
Mistake: B and D on the same side of the diagonal
The figure then crosses itself and is not the quadrilateral asked for.
Do this: In Case 1, place the two vertices on opposite sides of the diagonal.
Usually costs: Most of the marks.
Mistake: Rubbing out arcs
Arcs show the method.
Do this: Leave all construction arcs; draw them lightly but clearly.
Usually costs: 1 to 2 marks.
Mistake: Not checking the fourth angle
In Case 4, an error in one angle is caught by measuring ∠D.
Do this: Compute ∠D = 360° − (sum of the three) and measure to check.
Usually costs: Accuracy marks.
Mistake: Measuring sloppily
Answers usually need to be within about 1 mm or 1°.
Do this: Use a sharp pencil and read the protractor from the correct zero.
Usually costs: 1 mark.

Practice questions on Construction of Quadrilaterals

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQExam level
Which set of angles cannot be the angles of a quadrilateral?
(a) $90°, 90°, 90°, 90°$
(b) $100°, 110°, 90°, 70°$
(c) $60°, 120°, 60°, 120°$
(d) $75°, 105°, 85°, 95°$
Hint
Add each set.
Answer
(b): it adds to $370°$ .
Q2 · 1 mark · Very short answerExam level
In quadrilateral $A B C D$ , $∠ A = 75°$ , $∠ B = 80°$ and $∠ C = 120°$ . Find $∠ D$ .
Hint
360° minus the sum.
Answer
$85°$ .
Q3 · 2 marks · Short answerExam level
In Case 1 (four sides and diagonal $A C$ ), which triangles are constructed, and in what order?
Hint
Both triangles share $A C$ .
Answer
Triangle $A B C$ (from $A B$ , $B C$ , $A C$ ), then triangle $A C D$ (from $A D$ , $C D$ , $A C$ ) on the other side of $A C$ .
Q4 · 3 marks · Short answerHarder exam level
Construct parallelogram $A B C D$ with $A B = 6$ cm, $B C = 4$ cm and $∠ B = 60°$ . Measure $A C$ .
Hint
Opposite sides are equal: use arcs to find D.
Answer
Draw $A B$ , make $60°$ at $B$ , cut $B C = 4$ cm; arcs of 4 cm from $A$ and 6 cm from $C$ give $D$ . $A C \approx 5.3$ cm.
Q5 · 1 mark · Assertion-reasonHarder exam level
Assertion (A): A quadrilateral cannot be constructed uniquely from its four sides alone.
Reason (R): A square and a rhombus can have equal sides but different angles.
(a) (A) is true and (R) is false.
(b) (A) is false and (R) is true.
(c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
(d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).
Hint
Think of a hinged frame.
Answer
(c)
Q6 · 2 marks · Short answerHarder exam level
Construct a rectangle with sides 5 cm and 3.5 cm. Measure a diagonal.
Hint
Right angles at each corner.
Answer
Diagonal $\approx 6.1$ cm (both diagonals equal).
Q7 · 3 marks · Short answerHarder exam level
Construct a square whose diagonal is 6 cm. Measure its side.
Hint
Diagonals are equal and are perpendicular bisectors of each other.
Answer
Draw $A C = 6$ cm and its perpendicular bisector; mark $B$ , $D$ 3 cm from the mid-point. Side $\approx 4.2$ cm.
Q8 · 2 marks · Short answerHarder exam level
Is it possible to construct $A B C D$ with $A B = 5$ cm, $B C = 6$ cm, $C D = 4$ cm, $D A = 3$ cm and $B D = 9$ cm? Explain.
Hint
Check triangle $A B D$ .
Answer
No. In triangle $A B D$ , $5 + 3 = 8 < 9$ , so it cannot be drawn.
Q9 · 4 marks · Long answerHarder exam level
Construct $P QR S$ with $P Q = 4$ cm, $QR = 5$ cm, $R S = 5.5$ cm, $P R = 6.5$ cm and $QS = 6$ cm. Write the steps and measure $P S$ .
Hint
Three sides and two diagonals.
Answer
Triangle $P QR$ from $P Q$ , $QR$ , $P R$ ; then arcs of 5.5 cm from $R$ and 6 cm from $Q$ meet at $S$ across $P R$ from $Q$ . $P S \approx 3.4$ cm.
Q10 · 3 marks · Short answerHarder exam level
Construct $A B C D$ with $A B = 4.5$ cm, $B C = 5.5$ cm, $C D = 4$ cm, $∠ B = 60°$ , $∠ C = 90°$ . Measure $A D$ .
Hint
Three sides and two included angles.
Answer
$A D \approx 3.3$ cm.
Q11 · 4 marks · Long answerHOTS / competency
Construct $A B C D$ with $A B = 5$ cm, $B C = 6$ cm, $∠ A = 90°$ , $∠ B = 90°$ , $∠ C = 60°$ . Find $∠ D$ first, then measure $A D$ and $C D$ .
Hint
$∠ D = 360° - 240°$ .
Answer
$∠ D = 120°$ . Construction gives $A D \approx 3.1$ cm and $C D \approx 5.8$ cm.
Q12 · 4 marks · Long answerHOTS / competency
A rhombus has side 5 cm and one diagonal 8 cm. Explain how to construct it and find the other diagonal from your drawing.
Hint
The diagonal makes two triangles with sides 5, 5, 8.
Answer
Draw $A C = 8$ cm; arcs of 5 cm from $A$ and $C$ meet at $B$ and $D$ on opposite sides; join. $B D$ measures 6 cm (half-diagonals 4 cm and 3 cm with side 5 cm).

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.How many measurements fix a general quadrilateral?Show answer
5.
2.What should you draw before any construction?Show answer
A rough sketch with the measurements.
3.In Case 1, which line is drawn first?Show answer
The given diagonal.
4.In Case 3, which side is drawn first?Show answer
The middle side, between the two given angles.
5.What is the fourth angle if the other three are 80°, 100° and 90°?Show answer
90°.
6.What two measurements fix a rhombus?Show answer
Its two diagonals.
7.What is checked before drawing each triangle?Show answer
That any two sides add to more than the third.
8.Which angles can be made with compasses only?Show answer
60°, 120°, 90°, 30°, 45°, 75°, 105° and similar.

Best order to study it

Revise triangle constructions and constructing angles with compasses.
Practise Case 1 (four sides and a diagonal).
Practise Case 2 (three sides and two diagonals).
Practise Cases 3 and 4 (with angles).
Finish with special quadrilaterals and feasibility checks.

Quick revision checklist

The night before the exam: can you do each of these without looking?

Did you draw a rough sketch first?
Did you check the triangle condition for each triangle?
Did you draw the right line first for the case?
Are the vertices on the correct sides of the diagonal?
Are all arcs visible and vertices labelled?
Did you measure what the question asked for?

Construction of Quadrilaterals: frequently asked questions

Why can't four sides alone fix a quadrilateral?

Because the shape can change while the sides stay the same. A square can be pushed into a rhombus without changing any side.

Can I use a protractor for all angles?

You may use a protractor when the angle cannot be made with compasses, but where an angle such as 60° or 90° can be constructed, the exam may expect compasses.

What if my measured answer is slightly different from the expected one?

Small differences (about 1 mm or 1°) come from drawing. Larger differences mean an error, so check your arcs and angles.

Which triangle should I draw first?

One for which all three needed parts are given. In Case 1 that is either triangle on the diagonal; in Case 3 it is the side between the two angles.

Do I need to write the steps of construction?

Yes, usually. Number the steps clearly, in the order you carried them out.

Based on: CISCE Curriculum for Upper Primary Classes (VI-VIII), Mathematics (Class VIII); CISCE RDCD: Curriculum Pre-School to Class VIII