13.01 Pythagoras' theorem

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:

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.

Examples

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.

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

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+256	Evaluate the powers
	\displaystyle =	\displaystyle 337	Evaluate the sum

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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 324	Evaluate

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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.

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