TRANSFORMATION

2.1 Introduction

Transforming shapes into others of the same area can be applied in solving problems such as land disputes, allocation of plots and other design activities.

Did you know that a rectangular shape can be transformed Into a square shape but of equal area? Is it possible to transform a triangular shape into a square shape of equal area?

In design drawing, all this is possible through transformation procedures.

In this chapter, you will transform different shapes into other shapes of the same area.

2.2 Transforming Triangles and Quadrilaterals into other Shapes of Equal Area

Activity 2.1

Transforming a triangle Into a rectangle of equal area

What you need: 2H pencil, ruler, pair of compasses, dividers, eraser, T-square, board and clips

What to do: Individually, carry out this activity.

Follow the procedure given in Figure 2.3 and Figure 2.4 and transform the triangle into a rectangle of equal area.

Procedure

1. Draw the given triangle ABC, given that AB 70 mm, BC – 65 mm AC-55 mm.
2. 2. Drop a perpendicular from C to touch AB at D.

3. Bisect CD and prolong it on the left and right.

4. Drop perpendiculars from points A and B to meet this bisector at points E and D. Rectangle ABFE is the required one whose area is equal to that of the original triangle ABC. Proof:

i) Calculate the area of triangle ABC using the formula (1/2bh) in mm².

ii) Calculate the area of rectangle ABFE using the formula (LxW) in mm’.

iii) Compare the two areas.

Activity 2.2

Transforming a rectangle into a square of equal area

What you need: 2H pencil, ruler, pair of compasses, dividers, eraser, T-square,

board and clips

What to do: Individually, carry out the above activity.

Draw a rectangle and transform it into a square of equal area. Follow the procedure in Figure 2.7 to Figure 2.9

Procedure

1. Draw the given rectangle ABCD where AB = 80 mm and AD 50 mm and prolong base AB.

With centre B and radius BC, draw an arc to touch line AB at E.

3. Bisect AE to get centre O. Use radius OA and draw a semicircle AE. Prolong line BC to touch semicircle at F.

4. BF is one of the sides of the required square. Use it to complete the required square BHGF.

Activity 2.3

Transforming a quadrilateral into a triangle of equal area

What you need: 2H pencil, ruler, pair of compasses, dividers, eraser, T-square, board and clips

What to do: Individually, carry out this activity.

Draw the quadrilateral Figure 2.10 and transform it into a triangle of the same area. Follow the steps in Figure 2.11 and Figure 2.12.

1. Draw the given quadrilateral: AB=60 mm, BC= 50 mm, Angle at B=90°, CD-40 mm, AD=65 mm.
2. Prolong base AB to the right of B and join B to D. 3. Draw Line CE parallel to BD from C to touch extended base line at E.

4. Join B to E and also D to E.
5. Triangle AED is the required shape of equal area to the given quadrilateral ABCD.

Exercise 2.1

Transforming triangles and quadrilaterals to other shapes of equal areas In pairs, carry out this activity.

Apply the skills you have learnt to transform these triangles and quadrilaterals to other shapes with the same area.

1 Draw a right-angled triangle whose base is 40 mm and height of 66 mm. Transform it into a square of equal area.

2. Draw an isosceles triangle whose base is 46 mm and two base angles are 50 each. Transform it into a square of equal area.

3. Construct the parallelogram in Figure 2.13 and transform it into a rectangle of equal area.

2.3 Transforming Polygons

Transforming a pentagon into a square of equal area may not look to be an easy task at first sight. The only solution is to continue reducing its corners one by one until you arrive at the triangle. Do you still remember the procedure of transforming a triangle into a square of equal area?

Activity 2.4

4. Transform the triangle into a rectangle of equal area (Figure 2.19 and Figure 2.20).

5. Transform a rectangle into a square. use the transformed rectangle GFIH

Exercise 2.2

1. Draw the given irregular pentagon in Figure 2.22 and then transform it into a rectangle of equal area.
2. Draw a circle of diameter 60 mm and construct a regular hexagon in it. Using geometrical methods, demonstrate how to transform it into a triangle of equal area.

3. Draw a square of side 50 mm. Inscribe a regular octagon in it and then demonstrate how to transform it into an irregular pentagon of equal area.

2.4 Transformation, Calculation of Area and Cutting Out

In this sub topic, you will transform areas of varying shapes, carry out calculation of transformed areas to confirm their equality and then cut out the different shapes. You will use formulae to calculate the areas of various shapes. You will also need to use locally available materials to enable you trace and cut out the various shapes.

Activity 2.5

Transforming a parallelogram into a triangle of equal area

What you need: 2H pencil, ruler, pair of compasses, dividers, eraser, T-square, board, clips, cardboard paper, tracing paper, manilla paper, markers, paper pins, pair of scissors/knife/blade and locally available

What to do: Individually, carry out this activity.

Draw the parallelogram in Figure 2.23 and transform it into a triangle of equal area. Calculate and compare the areas of both the parallelogram and the triangle.

Procedure (Figure 2.24 and Figure 2.25)

1. Draw the parallelogram below.

2. Join BD and draw CE parallel to BD

3. Join D to E to obtain the required triangle AED with a vertical height of 40 mm.
4. Calculate and compare the areas. Trace out these two shapes on a manila paper or cardboard and cut them out. Present your work to the class and get feedback.

Exercise 2.3

In groups, carry out this activity.

1. Transform the parallelogram in Figure 2.26 into a square of equal area. Calculate and compare the areas of the two shapes. 2. Transform the triangle in Figure 2.27 into a parallelogram of equal area. Calculate and compare the two areas of the two shapes