Download Pythagorean triads

INVESTIGATION
STRAND: Measurement
CHAPTER: The right-angled triangle
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Pythagorean triads
LITERACY TASK • Find the meanings of these key terms:
1. Square number: ................................................................................................................................
2. Square root: ......................................................................................................................................
3. Hypotenuse: ......................................................................................................................................
4. Right-angle: .....................................................................................................................................
5. Factor: ............................................................................................................................................
6. Multiple: .........................................................................................................................................
Pythagoras’ theorem states that, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the
other two sides, a2 + b2 = c2.
The figure at right shows squares constructed on the three
sides (a, b and c) of a right-angled triangle.
1. Which side is the hypotenuse?
2. Measure the lengths of a, b and c in millimetres.
Calculate the areas of the squares on each of the
three sides.
3. What can you conclude about your findings in
question 2?
Since this set of three whole numbers (a, b and c) satisfies
Pythagoras’ theorem, it is called a Pythagorean triad or
a
c
triple.
4. On a separate sheet of paper, draw a rightangled triangle with side lengths half as long as
b
the triangle shown in the diagram previously.
As before, calculate the areas of each square
associated with the three sides. What do you
notice?
5. Repeat question 4 with a new right-angled triangle
whose side lengths are twice as long as the original
triangle.
6. Write a general statement concerning factors or
multiples of Pythagorean triads.
7. Draw a different sized right-angled triangle with side lengths of any measurement. Attach squares to each of the three sides and
calculate the areas of the three squares. Repeat steps 4 and 5, that is, investigate what happens when the side lengths are doubled
or halved. Do the side lengths and areas form the same relationship you discovered earlier? Comment on your results.
MQQLD2-pythagorean.pdf
© John Wiley & Sons Australia, Ltd 2004
STRAND: Measurement
CHAPTER: The right-angled triangle
INVESTIGATION: Pythagorean triads
Some rules have been developed to generate Pythagorean triads. Questions 8 and 9 are two examples of these. For these
questions, give two examples to illustrate each rule.
8. Start with an odd number greater than 1. Square the number then halve the answer. The two whole numbers either side of
this result, together with the original odd number, form a Pythagorean triad.
9. Start with an even number greater than 2. Square this number then divide the answer by 4. The two whole numbers either
side of this result, together with the original even number, form a Pythagorean triad.
MQQLD2-pythagorean.pdf
© John Wiley & Sons Australia, Ltd 2004