How to Use Pythagoras' Theorem to Find Missing Sides on Right-Angled Triangles
Missing sides on a right-angled triangle
Have you ever wondered how to find missing lengths on a right-angled triangle? Maybe you're faced with some maths homework. Maybe you're stuck on a real world problem, such as how high up a wall a ladder can reach. Either way, Pythagoras' theorem is generally the best tool to use when faced with a right-angled triangle with one missing side length. Let's take a look at how it works.
Squares on the sides of a right-angled triangle
What is Pythagoras' theorem?
Pythagoras' theorem states that for any right-angled triangle, the square on the hypotenuse (the longest of the three sides) is equal to the sum of the squares of the other two sides.
Put more simply, if we square the lengths of the two shorter sides and add the answers together, our final answer will equal the length of the hypotenuse squared. Algebraically, using a and b to denote the shorter sides and c to denote the hypotenuse as in the diagram below, this can be written as:
Put more simply, if we square the lengths of the two shorter sides and add the answers together, our final answer will equal the length of the hypotenuse squared. Algebraically, using a and b to denote the shorter sides and c to denote the hypotenuse as in the diagram below, this can be written as:
a^2 + b^2 = c^2
Who was Pythagoras?
Pythagoras' theorem takes its name from the ancient Greek mathematician and philosopher, Pythagoras of Samos (c 569 - 495 BC). Although little reliable evidence about Pythagoras' life remains, there are many mathematical and scientific discoveries attributed to him and his school of followers and pupils, the most famous of which is the theorem which bears his name.
Example 1: Finding the hypotenuse
For our first example, look at the triangle below which has shorter sides of length 5 cm and 12 cm, and a hypotenuse of unknown length, x cm. We want to find the length of the hypotenuse.
Using our equation a^2 + b^2 = c^2 we get:
x^2 = 5^2 + 12^2
= 25 + 144
= 169
Therefore x = square root 169 = 13 cm
It's always a good idea at this point to double-check that your answer makes sense. The hypotenuse is the longest of the three sides and so our answer should be bigger than both 5 and 12, which it is.
Also, because a straight line is the shortest distance between two points, any side of a triangle must be shorter than the sum of the other two sides. Again, this is the case in our example and so 13 cm is a sensible answer.
Using our equation a^2 + b^2 = c^2 we get:
x^2 = 5^2 + 12^2
= 25 + 144
= 169
Therefore x = square root 169 = 13 cm
It's always a good idea at this point to double-check that your answer makes sense. The hypotenuse is the longest of the three sides and so our answer should be bigger than both 5 and 12, which it is.
Also, because a straight line is the shortest distance between two points, any side of a triangle must be shorter than the sum of the other two sides. Again, this is the case in our example and so 13 cm is a sensible answer.
A right-angled triangle with missing hypotenuse measurement
Example 2: Finding a shorter side
Now, take a look at the image below. This time, we have a right-angled triangle with one of the shorter sides missing. As it is a right-angled triangle, we still use Pythagoras' theorem, but we need to be careful where we substitute our numbers.
In our equation a^2 + b^2 = c^2, c is the hypotenuse, so we must make sure to substitute the hypotenuse length, 20 cm, here while putting our unknown, p, on the left. So we get:
p^2 + 16^2 = 20^2
p^2 = 20^2 - 16^2
= 400 - 256
= 144
p = square root 144
= 12 cm
Again we check that the answer is sensible by making sure the hypotenuse is the biggest number and that no side is longer than the sum of the other two.
In our equation a^2 + b^2 = c^2, c is the hypotenuse, so we must make sure to substitute the hypotenuse length, 20 cm, here while putting our unknown, p, on the left. So we get:
p^2 + 16^2 = 20^2
p^2 = 20^2 - 16^2
= 400 - 256
= 144
p = square root 144
= 12 cm
Again we check that the answer is sensible by making sure the hypotenuse is the biggest number and that no side is longer than the sum of the other two.
A right-angled triangle with a missing shorter side
Using Pythagoras' theorem to find shorter sides
In the example above, we found the length of one of the shorter sides by substituting into the usual equation a^2 + b^2 = c^2 and then rearranging.
It is sometimes preferable to have rearranged the equation before substituting, in which case we get:
a^2 = c^2 - b^2
In short, when finding the hypotenuse we are adding the squares together and when finding a shorter side, we are taking the square of one shorter side away from the square of the hypotenuse.
It is sometimes preferable to have rearranged the equation before substituting, in which case we get:
a^2 = c^2 - b^2
In short, when finding the hypotenuse we are adding the squares together and when finding a shorter side, we are taking the square of one shorter side away from the square of the hypotenuse.
A real-world example of Pythagoras' theorem
A plane flies due east for 12 miles. It then changes direction and flies due north for 14 miles. How far from its starting point is it now?
By looking at the diagram below, we can see how Pythagoras' theorem can help us solve this problem.
By looking at the diagram below, we can see how Pythagoras' theorem can help us solve this problem.
The plane's journey
Solving the aeroplane problem
We can see from the diagram that when put with the eastwards and northwards journeys, the distance from the start point to the end, x, makes the hypotenuse of a right-angled triangle. We therefore get:
x^2 = 12^2 + 14^2
= 144 + 196
= 340
x = square root 340
= 18.44 miles (to 2 decimal places)
x^2 = 12^2 + 14^2
= 144 + 196
= 340
x = square root 340
= 18.44 miles (to 2 decimal places)
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