1.2 Why all the fuss about pythagoras?

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Why all this fuss about pythagoras?

INVESTIGATION

1.1 Work in a group but start on your own by drawing three right–angled triangles of different shapes and sizes. Work as accurately as possible. It will be a lot easier if you use squared paper. Now work even more accurately and measure the three sides of each triangle to the nearest millimetre. Complete the first three rows of the table. Now use your calculator to complete the rest of the table.

 SYMBOL TRIANGLE A TRIANGLE B TRIANGLE C Length of the shortest side a .................... .................... .................... Length of the medium triangle b .................... .................... .................... Length of the longest side c .................... .................... .................... Square of the length of the shortest side a 2 .................... .................... .................... Square of the length of the medium side b 2 .................... .................... .................... Sum of the two squares above a 2 + b 2 .................... .................... .................... Square of the length of the longest side c 2 .................... .................... ....................

1.2 There should be something interesting about the shaded cells of the table. In your group write down carefully what you notice and (if you can) why it happens.

2. Take three lines:

• The three given lines can be used to form a right–angled triangle.

Cutting–out:

2.1 Can the three given squares be used to form a triangle?

3 Write a neat summary of the results of the investigation.

end of INVESTIGATION

The Theorem of Pythagoras goes:

• In a right–angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.

The importance of the Theorem of Pythagoras is that we use it in two ways: Firstly, if we know that a triangle is right angled, then we can say something very important about its sides. Secondly, if we know that the three sides of a triangle have a certain relationship with each other, then we also know that the triangle must be right–angled.

CLASS WORK

1 We label triangles as follows:

• Refer to the sketch alongside.
• The three vertices (corners) get capital letters. (A, B and C).
• The sides can be named as the two vertices betweenwhich the side lies (AB, BC and AC), or we can uselowercase letters, each corresponding to the vertexopposite the side (a, b and c).
• At the moment we are dealing with right–angled triangles, but all triangles are labelled the same way.
• We also use the same letter to refer both to the name of a side and to its length.
• E.g. PR = 3,5cm or r = 5cm.ΔPRS means: triangle PRS

REMEMBER always to use a ruler for good sketches!

• In the following exercise the first problem is always an example.

2 Problem : AEH has a right angle at H.AH = 6cm and EH = 8cm.Draw a sketch (not accurate) of the triangle and use the Theorem of Pythagoras to calculate the length of side AE.

Solution : Because we know that the triangle has a rightangle, we are allowed to say that AE 2 = AH 2 + EH 2 (or: h 2 = e 2 + a 2 )

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