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T1 Pythagoras’ theorem

 

The Pythagorean theorem is a relationship between the lengths of the sides of a right angled triangle. It is useful for calculating side lengths in right triangles and is used in many parts of mathematics, science and engineering. It is one of those things you should know.

Right Angled Triangles

Triangles are plane shapes with three straight sides. A right angled triangle contains an angle of \(90^{\circ}\) as shown below.1 A right angle is an angle of \(90^{\circ}\).

a right angled triangle with the hypotenuse labelled

In a right angled triangle, the longest side is opposite the right angle and is called the hypotenuse.

Pythagoras’ Theorem

The theorem states that, in a right angled triangle,2 Pythagoras’ Theorem does not apply to any triangle - only right angled triangles. the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. In symbols:

a right angled triangle with the sides labelled a, b and h for hypotenuse

\[ h^{2}=a^{2}+b^{2}. \]

Pythagoras’ Theorem my be used to find the length of the third side of a triangle if you know the length of the other two sides.

Example 1

Find the length of the hypotenuse in the triangle below:

a right angled triangle with the shorter sides equal to 6 and 8 centimetres respectively

Solution: Using Pythagoras’ Theorem we have: \[\begin{align*} h^{2} & =a^{2}+b^{2}\\ & =6^{2}+8^{2}\\ & =36+64\\ & =100. \end{align*}\] So the hypotenuse \[\begin{align*} h & =\sqrt{100}\\ & =10\,. \end{align*}\] The length of the hypotenuse is \(10\,cm.\)3 Don’t forget to include the unit of measurement in your answer.

Example 2

Find the length of the side \(x\) to two decimal places in the triangle below:

a right angled triangle with the hypotenuse 20 centimetres long, one of the other sides 15 centimetres and the third side labelled x

Solution: In the case the hypotenuse \(h=20\:m\) and one of the shorter sides is \(15\,m\). Using Pythagoras’ Theorem we get: \[\begin{align*} 20^{2} & =15^{2}+x^{2}. \end{align*}\] Rearranging to make \(x^{2}\) the subject we have: \[\begin{align*} x^{2} & =20^{2}-15^{2}\\ & =400-225\\ & =175.\\ \textrm{Taking the square root of both sides gives:$\quad$ }\\ x & =\sqrt{175}\\ & =13.23\,. \end{align*}\] The length of the side \(x\) is \(13.23\,m\).

Example 3

Find the length of the unknown side in the triangle below to two decimal places:

a right angled triangle with the hypotenuse 12.5 centimetres, one of the shorter sides is 10.7 centimetres

Solution: In this case the hypotenuse \(h=12.5\,cm\) and one of the shorter sides is \(10.7\:cm\). Let the unknown side have length \(x\) then using Pythagoras’ Theorem: \[\begin{align*} 12.5^{2} & =10.7^{2}+x^{2}. \end{align*}\] Rearranging to make \(x^{2}\) the subject we have: \[\begin{align*} x^{2} & =12.5^{2}-10.7^{2}\\ & =156.25-114.49\\ & =41.76\,.\\ \textrm{Taking the square root of both sides gives:$\quad$ }\\ x & =\sqrt{41.76}\\ & =6.46\,. \end{align*}\] The length of the side is \(6.46\,cm.\)

Pythagorean Triples

In some right angled triangles, the length of all three sides is an integer.4 The integers is the set of whole numbers and is denoted by \[ \mathbb{Z}=\left\{ \cdots,-2,-1,0,1,2,\cdots\right\} . \] In this case the three side lengths are called a Pythagorean Triple. Some examples are \(\left(3,4,5\right)\), \(\left(5,12,13\right),\) \(\left(7,24,25\right)\) and \(\left(8,15,17\right)\). Multiples of these numbers are also Pythagorean triples. Multiplying the triple \(\left(3,4,5\right)\) by \(2\) gives a new triple \(\left(6,8,10\right)\) as we saw in Example 1 above.

Exercises

Find the missing side length in the following triangles.

a right angled triangle with the hypotenuse labelled as h and the other two sides of length 13 centimetres and 8 centimetres. The answer is 15.26 centimetres

a right angled triangle with hypotenuse 35 millimetres, another side 20 millimetres and the third side is labelled x. The answer is 28.72 millimetres

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