4.1 Compare quadrilaterals for similarities and differences  (Page 11/2)

Mathematics

Grade 9

Quadrilaterals, perspective drawing,transformations

Module 22

Compare quadrilaterals for similarities and differences

ACTIVITY 1

To compare quadrilaterals for similarities and differences

[LO 3.4]

1. Comparisons

For the next exercise you can form small groups. You are given pairs of quadrilaterals, which you have to compare. Write down in which ways they are alike and in which ways they are different. If you can say exactly by what process you can change the one into the other, then that will show that you have really understood them. For example, look at the question on parallel sides at the end of section 3 above.

Each group should work with at least one pair of shapes. When you work with a kite, you should consider both versions of the kite.

• Rhombus and square
• Trapezium and parallelogram
• Square and rectangle
• Kite and rhombus
• Parallelogram and kite
• Rectangle and trapezium

If, in addition, you would like to compare a different pair of quadrilaterals, please do so!

1. Definitions

A very short, but accurate, description of a quadrilateral using the following characteristics, is a definition . This definition is unambiguous, meaning that it applies to one shape and one shape only, and we can use it to distinguish between the different types of quadrilateral.

The definitions are given in a certain order because the later definitions refer to the previous definitions, to make them shorter and easier to understand. There is more than one set of definitions, and this is one of them.

• A quadrilateral is a plane (flat) figure bounded by four straight lines called sides.
• A kite is a quadrilateral with two pairs of equal adjacent sides.
• A trapezium is a quadrilateral with one pair of parallel opposite sides.
• A parallelogram is a quadrilateral with two pairs of parallel opposite sides.
• A rhombus is a parallelogram with equal adjacent sides.
• A square is a rhombus with four equal internal angles.
• A rectangle is a parallelogram with four equal internal angles.

ACTIVITY 2

To develop formulas for the area of quadrilaterals intuitively

[LO 3.4]

Calculating areas of plane shapes .

• Firstly, we will work with the areas of triangles. Most of you know the words “half base times height”. This is the formula for the area of a triangle, where we use A for the area , h for the height and b for the base .
• Area = ½ × base × height; A = ½ bh ; A = are various forms of the formula.
• But what is the base ? And what is the height ? The important point is that the height and the base make up a pair: the base is not any old side, and the height is not any old line.

• The height is a line that is perpendicular to the side that you choose as the base. Refer to the sketches above. The base and its corresponding height are drawn as darker lines. Below are three more examples showing the base/height pairs.

• Take two other colours, and in each of the above six triangles draw in the two other matching pairs of base/height, each pair in its own colour. Then do the following exercise:

Pick one of the triangles above, and calculate its area three times. Measure the lengths with your ruler, each time using another base/height pair. Do you find that answers agree closely? If they don’t, measure more carefully and try again.