Back to Edexcel Pythagoras’ Theorem (H) Home
4.9 A) Introduction & Finding the Hypotenuse
4.9 A) Introduction & Finding the Hypotenuse
We are able to use Pythagoras’ theorem to work out the unknown length of a right-angled triangle providing that we know the length of the other two sides. It is crucial that we only use Pythagoras’ theorem with right-angled triangles and not just any triangle. Below is an example of a right-angled triangle.
The longest side of the right-angled triangle is known as the hypotenuse. The hypotenuse is also the side that is opposite the right angle. The other two sides are shorter than the hypotenuse and they do not have specific names.
Pythagoras’ theorem states that:
Pythagoras’ theorem states that:
The square of the hypotenuse is equal to the squares of the other two sides added together
We can place Pythagoras’ theorem into algebraic form. In the algebraic form, c is the hypotenuse and a and b are the other two non-hypotenuse sides. The equation and the labelled up triangle is shown below.
The most classic example that you will come across when studying Pythagoras’ theorem is the 3, 4, 5 triangle. The sides of the 3, 4, 5 triangle are given below.
Let’s now use Pythagoras’ theorem to check that his equation does hold. His theorem is:
c in our equation is the length of the hypotenuse and the hypotenuse in the right-angled triangle above is 5 (the hypotenuse is the longest side and it is the side that is opposite the right-angle). The other two sides in the right-angled triangle are a and b. It does not matter which side we have as a and b; I am going to have a as 3 and b as 4. The triangle with the labelled sides are shown below.
We can now sub these values for a, b and c into the formula above.
We can see that Pythagoras’ theorem does hold.
Example 1
Let’s now have an example of a right-angled triangle where the length of the hypotenuse is not known.
What is the length of the hypotenuse in the following triangle?
Let’s now have an example of a right-angled triangle where the length of the hypotenuse is not known.
What is the length of the hypotenuse in the following triangle?
The question is asking us to find out what the length of the hypotenuse is. The word equation tells us that the square of the hypotenuse is equal to the square of the other two sides added together. The algebraic formula is given below.
The hypotenuse in this formula is c and the other two sides are a and b (it does not matter which side is a and which side is b). I am going to let a be the side that has a length of 5 and b be the side that has a length of 12. c is the hypotenuse, and this is the side that we are looking for. The labelled up triangle is shown below.
The next step is to sub the values for a and b into the formula.
We want to find the value of c and not c2. Therefore, we need to square root both sides of the equation.
The hypotenuse is 13 cm.
Example 2
What is the length of the hypotenuse in the triangle below? Give your answer to one decimal place.
What is the length of the hypotenuse in the triangle below? Give your answer to one decimal place.
We are finding out what the length of the hypotenuse is (we are looking for c). The square of the hypotenuse is equal to the square of the other two sides added together.
I am going to let a equal 7 and b equal 5 (it does not matter which side we label as a and b).
We want to find the value of c and not c2. Therefore, we need to square root both sides of the equation.
The final step is to round our answer to 1 decimal place.
Therefore, the hypotenuse for this triangle is 9.2 cm to 1 decimal place.