MATHEMATICS

Grade 9

QUADRILATERALS, PERSPECTIVE DRAWING,TRANSFORMATIONS

Module 21

EXPLORE AND IDENTIFY THE CHARACTERISTICS OF SOME QUADRILATERALS

ACTIVITY 1

To explore and identify the characteristics of some quadrilaterals

[LO 3.4]

In this work, you will learn more about some very important quadrilaterals. We need to know their characteristics as they occur often in the natural world, but especially in the manmade environment.

You will have to measure the lengths of lines and the sizes of angles, so you will need to have your ruler and protractor ready. For cutting out quadrilaterals you will need a pair of scissors.

First we start with the word quadrilateral. A quadrilateral is a flat shape with four straight sides, and, therefore four corners. We will study the sides (often in opposite pairs), the internal angles (also sometimes in opposite pairs), the diagonal lines and the lines of symmetry.

Look out for new words, and make sure that you understand their exact meaning before you continue.

1. Lines of symmetry

You have already encountered the quadrilateral we call a square.

The square

From your sheet of shapes, cut out the quadrilateral labelled “SQUARE”. Fold it carefully so that you can determine whether it has any lines of symmetry.

Lines of symmetry are lines along which any shape can be folded so that the two parts fall exactly over each other.

Make sure that you have found all the different lines of symmetry. Then mark the lines of symmetry as dotted lines on the sketch of the square alongside, using a ruler. One of them has been done as an example.

The dotted line in the sketch is also a diagonal, as it runs from one vertex (corner) to the opposite vertex.

- Look around you in the room. Can you find a square shape quickly?

If we push the square sideways, without changing its size, it turns into a rhombus.

1.2 The rhombus

Identify the RHOMBUS from the sheet of shapes. It is clear that it looks just like a square that is leaning over. Cut it out so that you can fold it to find its lines of symmetry.

Again, draw dotted lines of symmetry on this diagram

- Is the dotted line in this sketch a line of symmetry?

If we take a rhombus and stretch it sideways, then a parallelogram is produced.

1.3 The parallelogram

Find the PARALLELOGRAM on the sheet of shapes.

Cut it out so that you can fold it to find any lines of symmetry; draw them as dotted lines.

- You might have to search a bit to find something in the shape of a parallelogram. Your homework is to see whether you can find one in 24 hours.

This parallelogram turns into a rectangle when we push it upright.

1.4 The rectangle

Cut out the RECTANGLE and find its lines of symmetry to fill in on the rectangle alongside.

- Write down the differences you see between the rectangle and the square.

Now take the two end sides of the rectangle and turn them out in different directions to form a trapezium.

1.5 The trapezium

There is more than one TRAPEZIUM on the shape sheet. This is another example of a trapezium. Again, cut them out and find lines of symmetry.

- Using all the different kinds of trapezium as a guide, write down in words how you will recognise the shape.

1.6 On the shape sheet you will find two kinds of KITE. Cut out both kinds and find any lines of symmetry.

A kite is a kind of bird; it is also the name of the toy that can be made to fly in the wind, tethered by a string that is used to manipulate it. Modern kites have different ingenious shapes, but the quadrilateral gets its name from the simple paper kites, which are easy to make using two thin sticks of different lengths, some paper, glue and string – and a tail for a stabilizer.

Is there a special name for the dotted line in one of the kites above?

2. Side lengths

Study the examples of the six types of quadrilateral. First measure the sides of each as accurately as you can, to see whether any of the sides are the same length, and mark them. In this sketch of a parallelogram, the opposite sides have been marked with little lines to show which sides have equal lengths.

- Is a rhombus just a parallelogram with all four sides equal?

3. Parallel sides

Parallel lines (as you know) are lines that always stay equally far from each other. This means that they will never meet, no matter how far you extend them. They need not be the same length. You already know how to mark parallel lines with little arrows to show which are parallel.

Now study your quadrilaterals again to see whether you can identify the parallel lines with a bit of measuring. This is not easy, but you will do well if you concentrate and work methodically.

- If you could change just one side of any trapezium, could you turn it into a parallelogram? What would you have to change?

4. Internal angle sizes

It is easy to measure the internal angles with your protractor. Write the sizes in on the sketch, and then see whether you find right angles or equal angles. You can mark equal angles with lines to show which are which, as in this sketch of the parallelogram.

- Add up all the internal angles of every quadrilateral you measured and write the answer next to the quadrilateral. Does the answer surprise you?

5. Diagonals

Diagonals run from one internal vertex to the opposite vertex. Draw the diagonals in all the quadrilaterals (sometimes they will be on top of the lines of symmetry).

Measure the lengths of the diagonals to identify those quadrilaterals where the two diagonals are the same length. Mark them if they are the same, just as you marked the equal sides.

Use your protractor to carefully measure the two angles that the diagonals make where they cross (intersect). Take note of those quadrilaterals where the diagonals cross at right angles.

The diagonals also divide the internal angles of the quadrilateral. Measure these angles and make a note of those cases where the internal angle is bisected (halved) by the diagonal.

6. Tabulate your results

Complete the following table to summarise your results for all the characteristics of all the quadrilaterals.

Think very carefully about whether what you have observed is true for all versions of the same shape. For example, you may find that the two diagonals of a certain trapezium are equal; but would they be equal for all trapeziums? And if a kite has two equal diagonals, is it correct to call it a kite?

This table contains very useful information. Make sure your table is correct, and keep it for the following exercises.

Assessment

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Last edited by Siyavula Uploaders on Aug 13, 2009 6:47 am GMT-5.