Practical Geometry — Study Notes

1. Introduction

In the earlier chapters we studied geometry by looking at shapes that are already drawn. But very often in real life we need to construct shapes ourselves - architects draw plans, engineers mark plots, and carpenters make frames. The branch of geometry which deals with constructing figures using only a ruler and a compass is called practical geometry. These constructions are based on the properties of figures and on the fact that a set of given measurements uniquely determines a figure.

In this chapter we will learn how to construct quadrilaterals when different sets of measurements are given. A quadrilateral has ten parts in all - four sides, four angles and two diagonals. However, a quadrilateral is uniquely determined if we know any five of these parts in a suitable combination. We will construct quadrilaterals given (i) four sides and a diagonal, (ii) two diagonals and three sides, (iii) two adjacent sides and three angles, and (iv) three sides and two included angles.

2. Necessary Tools and Principles

Tools Needed

A ruler for measuring lengths and drawing straight lines, and a compass for drawing arcs and circles. A protractor is sometimes used for measuring angles, though constructions can be done with a compass alone.

Principle of Construction

A quadrilateral can be constructed uniquely when any of the following sets of measurements are given: 1. Four sides and one diagonal 2. Two diagonals and three sides 3. Two adjacent sides and three angles 4. Three sides and two included angles

Before constructing, always draw a rough sketch of the quadrilateral and mark the given measurements on it. The rough sketch helps in deciding the order of construction.

3. Construction of a Quadrilateral when Four Sides and One Diagonal are Given

When four sides and one diagonal are given, the diagonal divides the quadrilateral into two triangles. Since a triangle can be constructed uniquely if its three sides are known (SSS criterion), we construct both triangles using the given measurements.

Example

Construct a quadrilateral ABCD in which AB = 4.5 cm, BC = 5.5 cm, CD = 4 cm, AD = 6 cm and diagonal AC = 7 cm.

Steps: 1. Draw a rough sketch and mark the data. 2. Draw AC = 7 cm. 3. With A as centre and radius 4.5 cm, draw an arc. With C as centre and radius 5.5 cm, draw another arc. Let the arcs intersect at B. Join AB and BC. Triangle ABC is ready. 4. With A as centre and radius 6 cm, draw an arc. With C as centre and radius 4 cm, draw another arc. Let the arcs intersect at D. Join AD and CD. 5. Quadrilateral ABCD is complete.

4. Construction when Two Diagonals and Three Sides are Given

Here the two diagonals intersect at a point, and this helps us place the sides. The construction uses the fact that in some quadrilaterals, like a parallelogram, the diagonals bisect each other.

Example

Construct a parallelogram ABCD in which AB = 5 cm, BC = 3 cm and diagonal AC = 6 cm.

Since ABCD is a parallelogram, the diagonals bisect each other. Draw AC = 6 cm and mark its midpoint O. With O as centre and appropriate radius, arcs are drawn to locate B and D on opposite sides of AC.

5. Construction when Two Adjacent Sides and Three Angles are Given

When two adjacent sides and three angles are given, the three angles usually fix the directions of the sides. We use a protractor (or compass constructions) to draw the angles and locate the fourth vertex.

Example

Construct a quadrilateral ABCD in which AB = 5 cm, BC = 4 cm, angle A = 60 degrees, angle B = 110 degrees and angle C = 90 degrees.

Steps: 1. Draw AB = 5 cm. 2. At B, draw angle ABC = 110 degrees and cut BC = 4 cm along it. 3. At C, draw angle BCD = 90 degrees. 4. At A, draw angle DAB = 60 degrees. The two rays intersect at D. 5. Join AD and CD to get the required quadrilateral.

6. Construction when Three Sides and Two Included Angles are Given

When three sides and the two angles between them are given, we construct the triangle formed by the three sides and the two given angles, and then complete the quadrilateral.

Example

Construct a quadrilateral PQRS in which PQ = 4.5 cm, QR = 3.5 cm, RS = 5 cm, angle Q = 120 degrees and angle R = 90 degrees.

Steps: 1. Draw a rough sketch and mark the data. 2. Draw QR = 3.5 cm. 3. At Q, draw angle PQR = 120 degrees and cut PQ = 4.5 cm. 4. At R, draw angle QRS = 90 degrees and cut RS = 5 cm. 5. Join PS to complete the quadrilateral.

Quick Revision Tables

Table 1: Data Required for Unique Construction

Table 2: Common Measurements to Remember

Mind Map

Important Diagrams (SVG)

Diagram 1: Constructing a Quadrilateral ABCD with Four Sides and a Diagonal

Diagram 2: Constructing a Parallelogram with Two Diagonals and Three Sides

Common Mistakes

1. Students begin construction without drawing a rough sketch, which leads to wrong placement of vertices and confusion during construction.
2. Students do not maintain the exact opening of the compass while drawing arcs, which makes the measurements wrong.
3. While drawing an angle, students use the wrong scale of the protractor (inner or outer), getting a supplementary angle instead.
4. Students forget to mark the intersection points of arcs properly, or join the wrong intersection point to form the quadrilateral.
5. In the SSS construction, students may use the wrong side length for the wrong arc, making the triangle incorrect.
6. Students forget that in a parallelogram the diagonals bisect each other and must be marked at the midpoint, not at any other point.
7. After constructing, students often fail to check that all given measurements match the constructed figure, missing errors that could be corrected easily.

Exam Tips

1. Always begin every construction with a neat labelled rough sketch showing all the given measurements.
2. Write the given data clearly, for example "Given: AB = 4.5 cm, BC = 5.5 cm, ..." before starting.
3. Use a sharp pencil for accurate marks and keep the compass tight so arcs do not wobble.
4. Label all vertices after construction and mention "is the required quadrilateral" at the end.
5. For angle construction, remember the order of drawing the angles correctly and use the correct protractor scale.
6. Practise each of the four construction cases repeatedly, since construction questions in exams are formulaic and scoring.
7. Verify the final figure by measuring all sides and angles with the ruler and protractor before leaving the question.

Conclusion

Practical geometry transforms the theory of quadrilaterals into hands-on skill. In this chapter we learnt how to construct quadrilaterals using a ruler and compass when different combinations of measurements are given: four sides and a diagonal, two diagonals and three sides, two adjacent sides and three angles, and three sides with two included angles. We also understood why a rough sketch is essential, and how the properties of triangles (SSS criterion) and of parallelograms (bisecting diagonals) are used in constructions. These construction skills build accuracy, patience and geometric intuition, and they prepare students for advanced constructions of triangles, circles and tangents in higher classes.