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# 11. CONSTRUCTIONS

## 11.1 INTRODUCTION

In earlier chapters, the diagrams, which were necessary to prove a theorem or solving exercises were not necessarily precise. They were drawn only to give you a feeling for the situation and as an aid for proper reasoning. However, sometimes one needs an accurate figure, for example - to draw a map of a building to be constructed, to design tools, and various parts of a machine, to draw road maps etc. To draw such figures some basic geometrical instruments are needed. You must be having a geometry box which contains the following:

(i) A graduated scale, on one side of which centimetres and millimetres are marked off and on the other side inches and their parts are marked off.

(ii) A pair of set - squares, one with angles 90°, 60° and 30° and other with angles 90°, 45° and 45°.

(iii) A pair of dividers (or a divider) with adjustments.

(iv) A pair of compasses (or a compass) with provision of fitting a pencil at one end.

(v) A protractor.

Normally, all these instruments are needed in drawing a geometrical figure, such as a triangle, a circle, a quadrilateral, a polygon, etc. with given measurements. But a geometrical construction is the process of drawing a geometrical figure using only two instruments – an ungraduated ruler, also called a straight edge and a compass. In construction where measurements are also required, you may use a graduated scale and protractor also. In this chapter, some basic constructions will be considered. These will then be used to construct certain kinds of triangles.

## 11.1 BASIC CONSTRUCTIONS

In Class VI, you have learnt how to construct a circle, the perpendicular bisector of a line segment, angles of 30°, 45°, 60°, 90° and 120°, and the bisector of a given angle, without giving any justification for these constructions. In this section, you will construct some of these, with reasoning behind, why these constructions are valid.

## 11.3 SOME CONSTRUCTIONS OF TRIANGLES

So far, some basic constructions have been considered. Next, some constructions of triangles will be done by using the constructions given in earlier classes and given above. Recall from the Chapter 7 that SAS, SSS, ASA and RHS rules give the congruency of two triangles. Therefore, a triangle is unique if : (i) two sides and the included angle is given, (ii) three sides are given, (iii) two angles and the included side is given and, (iv) in a right triangle, hypotenuse and one side is given. You have already learnt how to construct such triangles in Class VII. Now, let us consider some more constructions of triangles. You may have noted that at least three parts of a triangle have to be given for constructing it but not all combinations of three parts are sufficient for the purpose. For example, if two sides and an angle (not the included angle) are given, then it is not always possible to construct such a triangle uniquely.

Construct a triangle ABC, in which ∠B = 60°, ∠C = 45° and AB + BC+ CA = 11 cm.

- Draw a line segment PQ = 11 cm.( = AB + BC + CA).
- At P construct an angle of 60° and at Q, an angle of 45°.
- Bisect these angles. Let the bisectors of these angles intersect at a point A.
- Draw perpendicular bisectors DE of AP to intersect PQ at B and FG of AQ to intersect PQ at C.
- Join AB and AC (see Fig. 11.9). Then, ABC is the required triangle.

**Fig 11.9**

In this chapter, you have done the following constructions using a ruler and a compass:

- To bisect a given angle.
- To draw the perpendicular bisector of a given line segment.
- To construct an angle of 60° etc.
- To construct a triangle given its base, a base angle and the sum of the other two sides.
- To construct a triangle given its base, a base angle and the difference of the other two sides.
- To construct a triangle given its perimeter and its two base angles.