2.2 Classifying and constructing triangles

Mathematics

Grade 8

Rational numbers, circles and triangles

Module 14

Classifying and constructing triangles

Activity 1

Classifying triangles, discovering important theorems about triangles and constructing triangles

[lo 3.1, 3.3, 3.4, 4.2.1]

• By the end of this learning unit, you will be able to do the following:

• understand how important the use of triangles is in everyday situations;
• explain how to find the unknown sides of a right-angled triangle (Pythagoras);
• calculate the area of a triangle;
• enjoy the action in geometry;
• use mathematical language to convey mathematical ideas, concepts, generalisations and mental processes.

1. When you classify triangles you can do it according to the angles or according to the sides.

1.1 Classification on the basis of the angles of a triangle:Are you able to complete the following?

a) Acute-angled triangles are triangles with

b) Right-angled triangles have

c) Obtuse-angled triangles have

1.2 Classification on the basis of the sides of the triangle:Are you able to complete the following?

a) An isosceles triangle has

b) An equilateral triangle has

c) A scalene triangle's

2. Are you able to complete the following theorems about triangles? Use a sketch to illustrate each of the theorems graphically.

THEOREM 1:

• The sum of the interior angles of any triangle is.........................

Sketch:

THEOREM 2:

• The exterior angle of a triangle is

Sketch:

3. Constructing triangles:

• Equipment: compasses, protractor, pencil and ruler

Remember this:

• Begin by drawing a rough sketch of the possible appearance.
• Begin by drawing the base line.

3.1 Construct $\Delta$ PQR with PQ = 7 cm, PR = 5 cm and $\hat{P}$ = 70°.

a) Sketch:

b) Measure the following:

1. QR = ........ 2. $\hat{R}$ = ........ 3. $\hat{Q}$ = ........ 4. $\hat{P}+\hat{Q}+\hat{R}=$ ........

3.2 Construct $\Delta$ KLM , an equilateral triangle. KM = 40 mm, KL = LM and $\hat{K}$ = 75°.Indicate the sizes of all the angles in your sketch.

Sketch:

Activity 2

Discovering the pythagorean theorem of pythagoras and calculating unknown sides with the help of this theorem

[lo 4.2.1, 4.8, 4.9, 4.10]

• The following could be done in groups.

Practical exercise: Making you own tangram.

1. Cut out a cardboard square (10 cm x 10 cm).

2. Draw both diagonals, because they form part of the bases of some figures.

3. Divide the square in such a way that the complete figure consists of the following:

3.1 two large equilateral triangles with bases of 10 cm in length;

3.2 two smaller equilateral triangles, each with base 5 cm in length;

3.3 one medium equilateral triangle with adjacent sides 5 cm in length;

3.4 one square with diagonals of 5cm;

3.5 one parallelogram with opposite sides of 5 cm.

• Make two of these. Cut along all the lines so that you will have two sets of the above shapes.

4. Now trace the largest triangle of your tangram in your workbook as a right-angled triangle.

5. Arrange the seven pieces to form a square and place this on the hypotenuse of the traced triangle.

6. Now arrange the two largest triangles to form a square and place this on one of the sides adja­cent to the right angle of the traced triangle.

7. Arrange the remaining pieces to form a square and place this on the other adjacent side.