Step by step
The question
Question: find the longest side of a right triangle with legs 7 and 8.
1 Name the sides
The longest side is called the hypotenuse.
We call it c.
It is across from the right angle, the biggest angle.
The two shorter sides make the right angle.
We call them a and b.
Either one can be a.
2 Squares on the sides
The rule for every right triangle: a² + b² = c².
Where does it come from?
Draw a square on each side.
a² means a times a.
That is the area of the square on side a.
b² is the area of the square on side b.
c² is the area of the big square, on the longest side.
In every right triangle, the two small squares together make the big square.
This was proved long ago, and it is true for every right triangle.
3 Put the numbers in
a is 7, so we write 7 in place of a.
b is 8, so we write 8 in place of b.
c stays c: it is the side we want.
4 Work out the squares
7² means 7 times 7.
That is 49.
8² means 8 times 8.
That is 64.
Write the squares: 49 + 64 = c².
49 plus 64 is 113.
The two small squares make 113.
So the big square is 113 too.
5 Take the square root
Now we have 113 = c².
c² means c times c.
But we want c alone, so we need to remove the ².
To remove a square, we take the square root: c is the square root of 113.
113 is not a square number: no whole number times itself makes 113.
So its root is a decimal that never ends.
10 × 10 is 100, too small.
11 × 11 is 121, too big.
So √113 is between 10 and 11.
With a calculator, √113 is about 10.6, to the nearest tenth.
So c is about 10.6.
6 Check
Let's check: put c = √113 back.
First a² + b².
Now c².
√113 times √113 is 113: that is what a square root means.
Both are 113.
So c = √113 is right!
Let's look back.
First: name the sides.
Then: squares on the sides.
Then: put the numbers in.
Then: work out the squares.
Then: take the square root.
Last: check.