Constructions — Study Notes

1. Introduction

Construction in geometry means drawing figures accurately using only two instruments: a compass and a straightedge (an unmarked ruler). No measurements with a marked ruler or protractor are allowed in the strictest sense, except for copying given lengths. The art of construction teaches us precision, neatness and logical sequencing, because each construction follows a fixed order of steps that must be performed correctly.

In earlier classes we learnt basic constructions like drawing perpendicular bisectors, angle bisectors and angles of known measure. In this chapter we will build on those skills. We will construct the perpendicular bisector of a line segment, the bisector of a given angle, and angles of 60, 90 and 45 degrees. We will also learn to divide a line segment in a given ratio using the internal bisector method and the parallel line method, and to construct triangles when different combinations of sides and angles are given.

2. Basic Constructions

Construction 1: The Perpendicular Bisector of a Line Segment

To draw the perpendicular bisector of segment AB: 1. Open the compass to a radius more than half of AB. 2. Draw arcs with this radius, centred at A, above and below the line. 3. Draw arcs with the same radius, centred at B, intersecting the earlier arcs. 4. Join the two intersection points. This line is the perpendicular bisector of AB, and its intersection with AB is the midpoint of AB.

Construction 2: The Bisector of a Given Angle

To bisect angle ABC: 1. With centre B and any convenient radius, draw an arc cutting BA and BC at P and Q. 2. With centre P and radius more than half of PQ, draw an arc. 3. With centre Q and the same radius, draw another arc intersecting the previous one at R. 4. Join B to R. The ray BR bisects angle ABC.

Construction 3: Angles of 60, 90 and 45 Degrees

• To construct a 60 degree angle: draw a ray, take any radius, draw an arc cutting the ray, and from that point draw another arc with the same radius; joining the vertex to the second intersection gives a 60 degree angle.
• To construct a 90 degree angle: construct a 60 degree angle and then bisect the remaining 120 degrees on the straight line, or draw the perpendicular to a line through a point.
• To construct a 45 degree angle: construct a 90 degree angle and then bisect it.

3. Dividing a Line Segment in a Given Ratio

Construction: Divide AB in the Ratio m:n

1. Draw a ray AX making an acute angle with AB.
2. Mark off m + n equal divisions on AX using the compass.
3. Join the last division point to B.
4. Through the m-th division point, draw a line parallel to the line joining the (m+n)-th point to B, using corresponding angles.
5. The point where this parallel line meets AB divides AB in the ratio m:n.

This construction uses the principle of the basic proportionality theorem (Thales theorem): a line parallel to one side of a triangle divides the other two sides in the same ratio.

4. Construction of Triangles

We can construct a triangle when the following data are given:

1. SSS (three sides): Draw one side, then draw arcs of the lengths of the other two sides from the endpoints, and join the intersection point to the endpoints.
2. SAS (two sides and the included angle): Draw the angle, mark the two sides on its arms, and join the endpoints.
3. ASA (two angles and the included side): Draw the base side, then draw the two given angles at its endpoints and extend their arms to meet.
4. RHS (right angle, hypotenuse and one side): Draw the right angle, mark the given side on one arm, and draw an arc of the hypotenuse length from its endpoint to meet the other arm.

Construction of a Triangle with a Given Base, Base Angle and the Sum of the Other Two Sides

A special construction exists when we are given the base BC, the base angle B, and the sum AB + AC of the other two sides. We construct the triangle as follows: 1. Draw the base BC and construct the given angle at B. 2. On the ray of this angle, mark a point D such that BD = AB + AC. 3. Join D to C and draw the perpendicular bisector of DC. 4. The perpendicular bisector meets the ray BD at A. Then AC = AD, so AB + AC = AB + AD = BD, and triangle ABC is the required triangle.

Construction of a Triangle with a Given Base, Base Angle and the Difference of the Other Two Sides

Similarly, if the difference AB - AC is given, we mark the difference on the ray from B and use the perpendicular bisector construction again. These constructions show how geometry theorems translate into practical drawing steps.

5. Constructions in Context

Constructions are not just an exercise in drawing; they are the practical application of the theorems we have proved. The perpendicular bisector construction uses the fact that the perpendicular bisector of a segment is the set of points equidistant from its endpoints. The angle bisector construction uses the property of points equidistant from the arms of the angle. The triangle constructions apply the congruence criteria, ensuring that the figure we draw is unique and correct. Practising these constructions also improves precision and builds confidence in using the compass and ruler accurately.

Quick Revision Tables

Table 1: Basic Constructions

Table 2: Triangle Constructions from Given Data

Mind Map

graph TD
    A["Constructions"] --> B["Basic constructions"]
    B --> C["Perpendicular bisector"]
    B --> D["Angle bisector"]
    B --> E["Angles 60, 90, 45 degrees"]
    A --> F["Division of line segment"]
    F --> G["Divide AB in ratio m:n using parallels"]
    A --> H["Triangle constructions"]
    H --> I["SSS, SAS, ASA, RHS"]
    H --> J["Base, base angle, sum or difference of sides"]
    A --> K["Tools"]
    K --> L["Compass and unmarked straightedge only"]

Important Diagrams (SVG)

Diagram 1: Construction of a Perpendicular Bisector

Diagram 2: Division of a Line Segment in a Given Ratio

Common Mistakes

1. Students often use a marked ruler to measure lengths during a construction. Constructions should use only the compass and an unmarked straightedge.
2. A very common error is using a radius less than or equal to half the segment when drawing the perpendicular bisector arcs; the arcs will not meet and the construction fails.
3. Students forget to show the construction lines and arcs in the final figure. The rough arcs must remain visible for the examiner to award full marks.
4. In dividing a line in the ratio m:n, students draw the parallel from the wrong division point, dividing the segment in the wrong ratio.
5. When constructing a triangle by SAS, students place the angle at the wrong end or draw the wrong side on the angle arm.
6. Students confuse the base angle construction for the sum of sides case, forgetting to mark AB + AC on the ray before drawing the perpendicular bisector.
7. A frequent mistake is to construct only part of the figure and leave out the final joining lines, producing an incomplete triangle.
8. Students do not write the steps of construction even when asked; writing clean numbered steps is part of the marks.

Exam Tips

1. Always write the steps of construction in the correct order before drawing the figure; many boards award separate marks for the steps.
2. Use a sharp pencil and keep the compass opening fixed while drawing arcs of the same radius.
3. Leave all construction arcs visible and do not erase the guiding lines, as they prove the correctness of your construction.
4. Practise the perpendicular bisector and angle bisector constructions daily, since they are the base of almost every other construction.
5. For dividing a segment in a ratio, always mark the divisions carefully and count them correctly before drawing the parallel.
6. In triangle constructions, first draw a rough sketch of the triangle and label it, then perform the exact construction beside it.
7. Mention the property or theorem used (for example, the basic proportionality theorem) when asked to justify the construction.

Conclusion

In this chapter we learnt the art of geometric construction using only a compass and a straightedge. We constructed the perpendicular bisector of a line segment, the bisector of a given angle, and angles of 60, 90 and 45 degrees. We learnt to divide a line segment in a given ratio using the basic proportionality theorem, and to construct triangles from various given data using the SSS, SAS, ASA and RHS criteria. We also studied the special constructions of triangles when the sum or difference of two sides is given along with the base and base angle. These constructions are not only examination questions themselves; they also reinforce the geometry theorems of the previous chapters and build the practical skills needed for higher-level geometry and drawing work.