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3.01 Pythagorean triples

Lesson

Pythagorean triples

The numbers 3,\,4 and 5 have a special property. If we square the first two numbers and add them they will be equal to the square of the largest number.\begin{aligned}3^{2}+4^{2} &= 5^{2}\\9+16&=25 \\25&=25\end{aligned}

We can also think about these numbers as satisfying the equation: a^{2}+b^{2}=c^{2}

That is if we substitute a with 3,\,b with 4, and c with 5 both sides will be equal. Any group of three whole numbers that satisfy the equation are called a Pythagorean triple or a Pythagorean triad. We can check any three numbers by substituting the two smaller numbers for a and b and the largest number for c.

Below are three Pythagorean triples: (3,\,4,\,5),\,\,(5,\,12,\,13),\,\,(8,\,15,\,17)

If you know one of the triples you can make another one by multiplying each number by a constant. For example if we multiply the triple (3,\,4,\,5) by 2 we will have (6,\,8,\,10).6^{2}+8^{2}=10^{2}

is also true so it will be a Pythagorean triple. The triples introduced above are now shown below with two of their multiples in each column:

(3,\,4,\,5)(5,\,12,\,13)(8,\,15,\,17)
\times 2(6,\,8,\,10)(10,\,24,\,26)(16,\,30,\,34)
\times 10(30,\,40,\,50)(50,\,120,\,130)(80,\,150,\,170)

The three numbers of a triple are often given from smallest to largest, however sometimes it may have the first two numbers swapped, and sometimes it is in no particular order. As long as you know which number is the largest you can check if three numbers are a Pythagoran triple.(3,\,4,\,5),\,\,(4,\,3,\,5),\,\,(5,\,3,\,4)

The biggest number is 5 in each case, this will be c.

A Pythagorean triple is any three whole numbers that satisfy a^{2}+b^{2}=c^{2} where c is the largest number.

Examples

Example 1

Is (5,\,12,\,13) a Pythagorean triple?

Worked Solution
Create a strategy

Use the formula: a^{2}+b^{2}=c^{2}.

Apply the idea

We have the given values for a=5,\, b=12 and c=13. Now we can find the values of each side of the formula.

\displaystyle a^{2}+b^{2}\displaystyle =\displaystyle 5^{2}+12^{2}Substitute a and b
\displaystyle =\displaystyle 25+144Evaluate the powers
\displaystyle =\displaystyle 169Evaluate
\displaystyle c^{2}\displaystyle =\displaystyle 13^{2}Substitute c
\displaystyle =\displaystyle 169Evaluate
\displaystyle =\displaystyle a^{2}+b^{2}Compare the two sides

So, the numbers (5,\,7,\,12) are a Pythagorean triple.

Reflect and check

We couldn't assume a^{2}+b^{2}=c^{2} was true. We had to test each side separately and then see if they were equal.

Idea summary

A Pythagorean triple is any three whole numbers that satisfy

\displaystyle a^{2}+b^{2}=c^{2}
\bm{c}
is the largest number in the triple
\bm{a}
is another number in the triple
\bm{b}
is another number in the triple

Right-angled triangles

A right angled triangle with hypotenuse and right angle labelled and opposite each other.

In a right-angled triangle the largest angle in the triangle is 90\degree. The side across from the right angle will be the largest side. We call this side the hypotenuse.

A right angled triangle with sides a and b and c on the hypotenuse. The Pythagoras theorem formula is written next to it.

All three sides of a right-angled triangle are related by the equation shown.

The two smaller sides will be called a and b, and the hypotenuse (the longest side) will be c.

Earlier we looked at Pythagorean triples which satisfy the same equation. Any triangle with sides that are a Pythagorean triple will be a right-angled triangle.

We can also test to see if a triangle is right-angled by checking to see if its three sides satisfy a^{2}+b^{2}=c^{2}.

Examples

Example 2

Use Pythagoras' theorem to determine whether this is a right-angled triangle.

A right angled triangle with side lengths of 9, 16, and 18.
a

Let a and b represent the two shorter side lengths. First find the value of a^{2}+b^{2}.

Worked Solution
Create a strategy

Substitute the lengths of the two shorter sides and add them.

Apply the idea
\displaystyle a^{2}+b^{2}\displaystyle =\displaystyle (9)^{2} + (16)^{2}Substitute the value of a and b.
\displaystyle =\displaystyle 81+256Evaluate
\displaystyle =\displaystyle 337Evaluate the sum
b

Let c represent the length of the longest side. Find the value of c^{2}.

Worked Solution
Create a strategy

Substitute the length of the longest side.

Apply the idea
\displaystyle c^{2}\displaystyle =\displaystyle (18)^{2}Substitute the value of c
\displaystyle =\displaystyle 324Evaluate
c

Is the triangle a right-angled triangle?

Worked Solution
Create a strategy

Compare the results from parts (a) and (b) to determine if a^2+b^2=c^2.

Apply the idea

337\neq 324

The triangle is not a right-angled triangle because a^2+b^2 \neq c^2.

Idea summary
A right angled triangle with sides a and b and c on the hypotenuse. The Pythagoras theorem formula is written next to it.

Pythagoras' theorem relates the three sides of a right-angled triangle, a and b are the two smaller sides, and the longest side, called the hypotenuse, is c.

We can also test to see if a triangle is right-angled by checking to see if its three sides satisfy a^{2}+b^{2}=c^{2}.

Outcomes

VCMMG318

Investigate Pythagoras’ Theorem and its application to solving simple problems involving right-angled triangles.

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