Step by step
The question
Question: find the longest side of a right triangle with legs 9 and 12.
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.
It is true for every right triangle.
3 Put the numbers in
a is 9, so we write 9 in place of a.
b is 12, so we write 12 in place of b.
c stays c: it is the side we want.
4 Work out the squares
9² means 9 times 9.
That is 81.
12² means 12 times 12.
That is 144.
Write the squares: 81 + 144 = c².
81 plus 144 is 225.
The two small squares make 225.
So the big square is 225 too.
5 Take the square root
Now we have c² = 225.
But we want c alone.
c² means c times c.
So we need to remove the ².
To remove a square, we take the square root.
So c is the square root of 225.
The square root of 225 is the plus number that times itself makes 225.
15 times 15 is 225.
−15 times −15 is 225 too, but the √ sign always means the plus one.
So c = 15.
The missing side is 15.
6 Check
Let's check: put c = 15 back.
First a² + b².
Now c².
Both are 225.
So c = 15 is right!
7 Type it in
The answer box takes 5 characters.
15 has 2.
It fits.
We type 15.
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.
Then: check.
Last: type it in.