Pythagoras’ Theorem PYTHAGOREAN TRIPLES. What are Pythagorean Triples?  Aka Pythagorean Triads  Are 3 whole numbers that satisfy Pythagoras’ Theorem.

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Presentation transcript:

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 c 2 = a 2 + b 2 2. Substitute the numbers 10 2 = Square all the numbers 100 = Add the numbers on the RHS 100 = Answer the question in a mathematical sentence c 2 ≠ a 2 + b 2

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 = S 2 – 1, where S=shortest side and M=middle side 2  These two numbers are then used to find the third side (hypotenuse

1. Write the equation M = S 2 – Substitute in the smallest number as ‘s’ M = 7 2 – 1 2 Example 3: If the smallest number in a Pythagorean Triple is 7, find the middle number and, hence, find the third number 3. Square the smallest number M = 49 – Solve the equation M = 48 = Use Pythagoras’ Theorem to find the hypotenuse c 2 = a 2 + b 2 c 2 = c 2 = c 2 = 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: 2 x y, x 2 – y 2, and x 2 + y 2

Example 4: Obtain a Pythagorean Triple using the values x =7 and y=2 using Method 3 1. Write the first equation 2 x y 2. Substitute in the values 2 x 7 x 2 3. Solve the equation Write the second equation x 2 – y 2 5. Substitute in the values 7 2 – Solve the equation 49 – 4 = Write the third equation x 2 + y 2 8. Substitute in the values Solve the equation = 54

Your turn! Exercise 6D page 183 Level 1 Q 1 2 Complete composite shape worksheet (click here)(click here Level 2 Qs