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Pythagoras’ Theorem
PYTHAGOREAN TRIPLES
What are Pythagorean Triples?

Aka Pythagorean Triads

Are 3 whole numbers that satisfy Pythagoras’ Theorem

That is, 3 whole numbers that could be the sides of a right-angled triangle
Method 1

Used when you know the lengths of all 3 sides

The LHS of the equation must equal the RHS of the equation
Example: Do the numbers 5, 7 and 10 make a Pythagorean Triple?
1. Write the equation
c2 = a2 + b2
2. Substitute the numbers
102 = 52 + 72
3. Square all the numbers
100 = 25 + 49
4. Add the numbers on the RHS
100 = 94
5. Answer the question in a mathematical sentence
c2 ≠ a2 + b2
Example 2: Do the numbers 3, 4 and 5 make
a Pythagorean Triple?
Method 2

Used when you know only one side, and the measurement is a odd number

Uses the formula M = S2 – 1, where S=shortest side and M=middle side
2

These two numbers are then used to find the third side (hypotenuse
Example 3: If the smallest number in a Pythagorean
Triple is 7, find the middle number and, hence, find
the third number
1. Write the equation
M = S2 – 1
2
2. Substitute in the smallest
number as ‘s’
M = 72 – 1
2
3. Square the smallest
number
M = 49 – 1
2
4. Solve the equation
M = 48 = 24
2
5. Use Pythagoras’
Theorem to find the
hypotenuse
c2 = a2 + b2
c2 = 72 + 242
c2 = 49 + 576
c2 = 625
c = √625
c = 25
Method 3

Used when you are given an x and y value

Numbers must be whole numbers and the
x value must be larger than the
y value

The triple is given by finding: 2xy, x2 – y2, and x2 + y2
Example 4: Obtain a Pythagorean Triple
using the values x=7 and y=2 using Method 3
1. Write the first equation
2xy
2. Substitute in the values
2x7x2
3. Solve the equation
28
4. Write the second equation
x2 – y2
5. Substitute in the values
72 – 22
6. Solve the equation
49 – 4 = 45
7. Write the third equation
x2 + y2
8. Substitute in the values
72 + 22
9. Solve the equation
49 + 4 = 54
Your turn!
Exercise 6D page 183
Level 1
• Q1
2
• Complete
composite shape
worksheet (click
here)
Level 2
Qs 1
2
3
5
8