21

Constructions

21.1 INTRODUCTION

One of the aims of the studying Geometry is to acquire the skill of drawing figures accurately. You have learnt how to construct geometrical figures namely triangles, squares and circles with the help of ruler and compasses. You have constructed angles of 30°, 60°, 90°, 120° and 45°. You have also drawn perpendicular bisector of a line segment and bisector of an angle. In this lesson we will extend our learning to construct some other important geometrical figures. 21.2 OBJECTIVES

After studying this lesson, the learner will be able to :

z divide a given line segment internally in a given ratio.

z Construct a triangle from the given data

(i) SSS

(ii) SAS

(iii) ASA

(iv) RHS

(v) perimeter and base angles

(vi) base, sum/difference of the other two sides and one base angle. (vii) two sides and a median corresponding to one of these sides. z Construct rectilinear figures such as parallelograms, rectangles, squares, rhombuses and trapeziums.

z Construct a quadrilateral from the given data

(i) four sides and a diagonal

(ii) three sides and both diagonals

(iii) two adjacent sides and three angles158 Mathematics

(iv) three sides and two included angles

(v) four sides and an angle.

z Construct a triangle equal in area to a given quadrilateral. z Construct tangents to a circle from a point

(i) outside it

(ii) on it using the centre of the circle

z Construct circumcircle of a triangle

z Construct incircle of a triangle.

21.3 EXPECTED BACKGROUND KNOWLEDGE

We assume that the learner already knows how to use a pair of compasses and ruler to construct z angles of 30°, 45°, 60°, 90°, 105°, 120°.

z the right bisector of a line segment

z bisector of a given angle.

z parallelograms, rhombuses, rectangles, and squares

z a circle

21.4 DIVISION OF A LINE SEGMENT IN THE GIVEN RATIO INTERNALLY Construction 1 : To divide a line segment internally in a given ratio. Given a line segment AB. You are required to divide it internally in the ratio 2 : 3. We go through the following steps.

Step 1 : Draw a ray AC making an acute angle with AB.

Step 2 : Starting with A, mark off 5 points C1, C2, C3, C4 and C5 at equal distances from the point A.

Step 3 : Join C5 and B.

Step 4 : Through C2 (i.e. the second point), draw C2D parallel to C5B meeting AB in D. Fig. 21.1

Then D is the required point which divides AB internally in the ratio 2 : 3 as shown in Fig. 21.1Constructions 159

CHECK YOUR PROGRESS 21.1

1. Draw a line segment 7 cm long. Divide it internally in the ratio 3 : 4. Measure each part. Also write the steps of construction.

2. Draw a line segment PQ = 8 cm. Find the point R on it such that PR = 3

4

PQ .

[Hint : Divide the line segment PQ internally in the ratio 3 : 1]. 21.5 CONSTRUCTION OF TRIANGLES

Construction 2 : To construct a triangle when three sides are given (SSS) Suppose you are required to construct ∆ABC in which AB = 6 cm, AC = 4.8 cm and BC = 5 cm.

We go through the following steps :

Step 1 : Draw AB = 6 cm.

Step 2 : With A as centre and radius 4.8 cm, draw an arc.

Step 3 : With B as centre and radius 5 cm draw another arc

intersecting the arc of Step 2 at C.

Step 4 : Join AC and BC.

Then ∆ABC is the required triangle.

[Note : You may take BC or AC as the base]

Construction 3 : To construct a triangle, when two sides and the included angle is given (SAS) Suppose you are required to construct a triangle PQR in which PQ = 5.6 cm, QR = 4.5 cm and ∠PQR = 60°

For constructing the triangle, we go through the following steps. Step 1 : Draw PQ = 5.6 cm

Step 2 : At Q, construct an angle ∠PQX = 60°

Step 3 : With Q as centre and radius 4.5 cm draw an arc

cutting QX at R.

Step 4 : Join PR

Then ∆PQR is the required triangle.

[Note : You may take QR = 4.5 cm as the base instead of PQ]

Construction 4. To construct a triangle when two angles and the included side are given (ASA). Let us construct a ∆ABC in which ∠B = 60°, ∠C = 45° and BC =...