Pythagoras 1
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Introduction to the Rule of Pythagoras.

This topic is part of the TCS FREE high school mathematics 'How-to Library'. It introduces you to the rule of Pythagoras for right triangles, as well as Pythagorian triples.
(See the index page for a list of all available topics in the library.) To make best use of this topic, you need to download the Maths Helper Plus software. Click here for instructions.

Theory:

The rule of Pythagoras applies to right angled triangles. It can be used to find an unknown side of a right angled triangle, or to prove that a given triangle is right angled.

 A right angled triangle, or ‘right’ triangle, contains a right angle:

The longest side in a right angled triangle is called the ‘hypotenuse’. The hypotenuse is always opposite to the right angle. The two other sides are shorter than the hypotenuse, and are called ‘legs’.

The rule of Pythagoras states that in any right angled triangle with legs ‘a’ and ‘b’ and hypotenuse ‘c’...

 

 NOTE: It follows that for any triangle with side lengths ‘a’, ‘b’ and ‘c’, if c2 = a2 + b2, then that triangle is a right angled triangle.

 Demonstrating the rule of Pythagoras using areas.

 To demonstrate the rule of Pythagoras we will use the fact  that ‘a2’ is the area of a square with side ‘a’, and similarly for ‘b’ and ‘c. This diagram shows the sides and areas:

 

It follows from the rule of Pythagoras that: area3 = area1 + area2

 Method:

Maths Helper Plus can demonstrate the rule of Pythagoras using areas. Follow the steps below to demonstrate the rule, and also calculate the length of the hypotenuse, 'C'. 

Step 1 Download the free support file... We have created a Maths Helper Plus document containing the completed example from this topic. You can use this to practice the steps described below, and as a starting point for solving your own problems.

File name:  'Pythagoras 1.mhp'   File size: 9kb
Click here to download the file.

If you choose 'Open this file from its current location', then Maths Helper Plus should open the document immediately. If not, try the other option: 'Save this file to disk', then run Maths Helper Plus and choose the 'Open' command from the 'File' menu. Locate the saved file and open it. If you do not yet have Maths Helper Plus installed on your computer, click here for instructions.

 

Step 2  Display the parameters box

Press the F5 key to display the parameters box:

 

You enter the given information into these edit boxes as follows:

 

        edit box 'A' = 'a', the horizontal leg of the triangle

        edit box 'B' = 'b', the vertical leg of the triangle

 

Set  A, B to the required leg lengths, then click the 'Update' button to refresh the diagram and calculate the hypotenuse, 'c'.

NOTE: To vary 'A' or 'B' gradually as an animation, first click with the mouse on the edit box for A or B, then click on the 'slider'. (See diagram above). Now you can use the up and down arrow keys on your keyboard to animate the diagram.

A) Testing the rule of Pythagoras:

Here are some values of 'a' and 'b' to enter into Maths Helper Plus.
Print this page, then first fill in the blanks for all but the c2 values in the last column below. Then enter the 'a' and 'b' values into Math Helper Plus to calculate the c2 values.

HINT: If the diagram becomes too big, press F10 to make it smaller. If it is too small, Shift+F10 makes it larger.

Now write the c2 values in the last column.  Does  c2 = a2 + b2 ?

1.  a = 3, a2 = _____, b = 4, b2 = ______, a2 + b2 = _____,  c2 = _____
2.  a = 2, a2 = _____, b = 4, b2 = ______, a2 + b2 = _____,  c2 = _____
3.  a = 5, a2 = _____, b = 3, b2 = ______, a2 + b2 = _____,  c2 = _____
4.  a = 5, a2 = _____, b = 6, b2 = ______, a2 + b2 = _____,  c2 = _____
5.  a = 6, a2 = _____, b = 1, b2 = ______, a2 + b2 = _____,  c2 = _____

B) Pythagorean Triples:

A set of three integers: { a, b, c } is a ‘Pythagorean triple’ if  a2 + b2 = c2. Three integers will only form the lengths of the sides of a right angled triangle if they are a Pythagorean triple.

For example, { 3, 4, 5 } is a Pythagorean triple because 32 + 42 = 52 , so we can make a right angled triangle with these side lengths.

 To see if three numbers are a Pythagorean triple:

(a) The two smallest numbers are ‘a’ and ‘b’. Find the sum of the squares of the smallest two of the three numbers: a2 + b2

(b) The largest number is ‘c’. Calculate c2.

(c) If  a2 + b2 = c2  then the integers:  { a, b, c } form a Pythagorean triple.

For each set of three integers below, find out if they form a Pythagorean triple.

1. { 3, 4, 5 }

2. { 2, 3, 4 }

3. { 13, 5, 12 }

4. { 24, 25, 7 }

5. { 11, 9, 4 }

6. { 8, 15, 17 }

Use Maths Helper Plus to correct your work, like this...

(a) Enter the first two numbers into Maths Helper Plus as 'A' and 'B'.

(b) Click 'Update' to calculate the 'c' value using the rule of Pythagoras.

(c) If the 'c' value calculated by Maths Helper Plus is the same as the third number in the set, then the three numbers form a Pythagorian Triple.

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