Step by step
The question
Question: find the missing side of a right triangle with leg 5 m and longest side 13 m.
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
13 is the longest side, so it is c.
a is 5, so we write 5 in place of a.
c is 13, so we write 13 in place of c.
b stays b: it is the side we want.
4 Work out the squares
5² means 5 times 5.
That is 25.
13² means 13 times 13.
That is 169.
Write the squares: 25 + b² = 169.
Is b alone now?
Not yet: + 25 is still with b.
How do we remove it?
5 Remove the 25
We want b² by itself.
The 25 is in the way.
To remove 25, we take away 25.
We do it on both sides.
Why both sides?
The = sign means both sides are the same amount.
If we change only one side, they are not the same any more.
25 and − 25 cancel each other.
They make zero.
169 minus 25 is 144.
So the square on side b is 144.
So after step 5: we took away 25, and now we have b² = 144.
6 Take the square root
Now we have b² = 144.
But we want b alone.
b² means b times b.
So we need to remove the ².
To remove a square, we take the square root.
So b is the square root of 144.
The square root of 144 is the plus number that times itself makes 144.
12 times 12 is 144.
−12 times −12 is 144 too, but the √ sign always means the plus one.
So b = 12 m.
The missing side is 12 m.
7 Check
Let's check: put b = 12 m back.
First a² + b².
Now c².
Both are 169.
So b = 12 m is right!
8 Type it in
The answer box takes 5 characters.
12 has 2.
It fits.
We type 12.
Let's look back.
First: name the sides.
Then: squares on the sides.
Then: put the numbers in.
Then: work out the squares.
Then: remove the 25.
Then: take the square root.
Then: check.
Last: type it in.