Step by step
The question
Question: find the longest side of a right triangle with legs 5 and 9.
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 5, so we write 5 in place of a.
b is 9, so we write 9 in place of b.
c stays c: it is the side we want.
4 Work out the squares
5² means 5 times 5.
That is 25.
9² means 9 times 9.
That is 81.
Write the squares: 25 + 81 = c².
25 plus 81 is 106.
The two small squares make 106.
So the big square is 106 too.
5 Take the square root
Now we have 106 = 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 106.
106 is not a square number: no whole number times itself makes 106.
So its root is a decimal that never ends.
10 × 10 is 100, too small.
11 × 11 is 121, too big.
So √106 is between 10 and 11.
With a calculator, √106 is about 10.3, to the nearest tenth.
So c is about 10.3.
6 Check
Let's check: put c = √106 back.
First a² + b².
Now c².
√106 times √106 is 106: that is what a square root means.
Both are 106.
So c = √106 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.