ICSE Class 8 Maths Geometry Chapter 29 Construction of Quadrilaterals

Construction Of Quadrilaterals

Construction of Quadrilaterals

This chapter deals with the steps of construction of various types of quadrilaterals. It is advisable to draw a rough sketch of the required quadrilaterals to help plan the constructions.

I. Given the measure of 4 sides and 1 diagonal

Construct quadrilateral PQRS, given PQ = 5 cm, QR = 5.5 cm, RS = 6.5 cm, SP = 4.5 cm, and PR = 7.5 cm.

Sketch

The sketch shows a quadrilateral with the following measurements: PQ = 5, QR = 5.5, RS = 6.5, SP = 4.5, and diagonal PR = 7.5.

Construction Steps

1. Draw PQ = 5 cm.

2. With Q as centre and radius 5.5 cm, draw arc q.

3. With P as centre and radius 7.5 cm, draw arc p that intersects arc q at point R.

4. With P as centre and radius 4.5 cm, draw arc p₁.

5. With R as centre and radius 6.5 cm, draw arc r that intersects arc p₁ at point S.

6. Connect point S with P and R and point R with P and Q.

7. We have quadrilateral PQRS (Figure 29.1), where PQ = 5 cm, QR = 5.5 cm, RS = 6.5 cm, SP = 4.5 cm, and diagonal PR = 7.5 cm.

II. Given the measure of 4 sides and 1 angle

Construct quadrilateral PQRS, given PQ = 5 cm, QR = 4 cm, RS = 4.5 cm, SP = 3.5 cm, and \(\angle SPQ = 60°\).

Sketch

The sketch shows a quadrilateral with the following measurements: PQ = 5, QR = 4, RS = 4.5, SP = 3.5, and angle SPQ = 60°.

Construction Steps

1. Draw PQ = 5 cm.

2. At point P construct \(\angle APQ = 60°\).

3. With P as centre and radius 3.5 cm, draw arc p that intersects arc p at point S.

4. With Q as centre and radius 4 cm, draw arc q.

5. With S as centre and radius 4.5 cm, draw arc s that intersects arc q at point R.

6. Connect point S with R and connect point R with Q.

7. We have quadrilateral PQRS (Figure 29.2), where PQ = 5 cm, QR = 4 cm, RS = 4.5 cm, SP = 3.5 cm, and \(\angle SPQ = 60°\).

III. Given 3 consecutive sides and the included angles

Construct quadrilateral PQRS, given PQ = 2.6 cm, QR = 4 cm, RS = 2.5 cm, \(\angle PQR = 45°\), and \(\angle QRS = 60°\).

Sketch

The sketch shows measurements: PQ = 2.6, angle PQR = 45°, QR = 4, angle QRS = 60°, RS = 2.5.

Construction Steps

1. Draw QR = 4 cm.

2. At point Q construct \(\angle AQR = 45°\) and at point R construct \(\angle QRB = 60°\).

3. With R as centre and radius 2.5 cm, draw arc r that intersects ray RB at point S.

4. With Q as centre and radius PQ = 2.6 cm, draw arc q that intersects ray QA at point P.

5. Connect points P and S.

6. We have quadrilateral PQRS (Figure 29.3), where PQ = 2.6 cm, QR = 4 cm, RS = 2.5 cm, \(\angle PQR = 45°\), and \(\angle QRS = 60°\).

Teacher's Note

Understanding quadrilateral construction helps students visualize how geometric shapes are built systematically, similar to how architects design buildings using precise measurements and angles.

Construction of Parallelograms

I. Given 2 consecutive sides and 1 angle

Construct parallelogram PQRS, given PQ = 3.5 cm, QR = 4 cm, and \(\angle QRS = 45°\).

Sketch

The sketch shows measurements: PQ = 3.5, QR = 4, angle QRS = 45°.

From the above sketch we can see that the construction would be easier if the included angle is found.

\(\angle PQR = 180° - 45° = 135°\)

Construction Steps

1. Draw QR = 4 cm.

2. At point Q construct \(\angle AQR = 135°\).

3. With Q as centre and radius 3.5 cm, draw arc q that intersects ray QA at point P.

4. With R as centre and radius 3.5 cm, draw arc r.

5. With P as centre and radius 4 cm, draw arc p that intersects arc r at point S.

6. Connect point S with points P and R.

7. We have parallelogram PQRS (Figure 29.4), where PQ = 3.5 cm, QR = 4 cm, and \(\angle QRS = 45°\).

II. Given 2 adjacent sides and a diagonal

Construct parallelogram PQRS, given PQ = 5 cm, QR = 3 cm, and PR = 7 cm.

Construction Steps

1. Draw PQ = 5 cm.

2. With P as centre, measure off diagonal PR = 7 cm with arc p.

3. With Q as centre and radius 3 cm, draw arc q that intersects arc p at point R.

4. With P as centre and radius 3 cm, draw arc p₁.

5. With R as centre and radius 5 cm, draw arc r that intersects arc p₁ at point S.

6. Connect point S with points P and R and point R with points P and Q.

7. We have parallelogram PQRS (Figure 29.5), where PQ = 5 cm, QR = 3 cm, and PR = 7 cm.

Teacher's Note

Parallelograms appear in real life as window panes, book covers, and floor tiles, making this construction technique practically valuable for design work.

Construction of Rectangles

I. Given 2 adjacent sides

If the measures of two adjacent sides of a rectangle are given, we know that the angle included between them is 90°. Thus, we follow the steps described earlier in construction I of parallelograms.

II. Given 1 side and 1 diagonal

Construct rectangle PQRS, given PQ = 4 cm and PR = 5 cm.

Construction Steps

1. Draw PQ = 4 cm.

2. At point Q construct \(\angle PQA = 90°\).

3. With P as centre, measure off diagonal PR = 5 cm with arc p that intersects ray QA at point R.

4. With R as centre and radius 4 cm, draw arc r.

5. With P as centre, and a radius equal to line segment QR, measure off SP with arc p₁ that intersects arc r at point S.

6. Connect point S with points P and R.

7. We have rectangle PQRS (Figure 29.8), where PQ = 4 cm and diagonal PR = 5 cm.

III. Given 1 diagonal and the angle between the diagonals

A rectangle being a special type of parallelogram in which the diagonals are equal in measure, given the measure of 1 diagonal and the angle between the diagonals, we follow the steps described earlier in construction IV of parallelograms.

Teacher's Note

Rectangles are the most common shapes in construction and design, from doors and windows to computer screens, making this knowledge essential for practical applications.

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