Step by step
The question
Question: find the missing side of a right triangle with leg 4 and longest side 8.
1 Name the sides
The longest side is called the hypotenuse.
We call it c.
It is always across from the right angle.
The two shorter sides make the right angle.
We call them a and b.
The rule for every right triangle: a² + b² = c².
2 Put the numbers in
Now put the numbers in: a = 4 and c = 8.
3 Work out the squares
4² means 4 times 4.
That is 16.
8² means 8 times 8.
That is 64.
Write the squares: 16 + b² = 64.
4 Remove the 16
We want b² by itself.
The 16 is in the way.
To remove 16, we take away 16.
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.
16 and − 16 make zero.
Cross them out.
64 minus 16 is 48.
5 Take the square root
b² is 48.
So b times b is 48.
That means b is the square root of 48.
48 is not a square number.
It is between 6 and 7.
But 48 has a square number in it: 16.
48 is 16 times 3.
A root of a times splits into two roots.
√16 is 4.
So b = 4√3.
That is the exact answer, made simple.
With a calculator, 4√3 is about 6.9, to the nearest tenth.
So b is about 6.9.
6 Check
Let's check: put b = 4√3 back.
First a² + b².
(4√3)² is 4² times (√3)²: 16 times 3.
Now c².
Both are 64.
So b = 4√3 is right!
Let's look back.
First: name the sides.
Then: put the numbers in.
Then: work out the squares.
Then: remove the 16.
Then: take the square root.
Last: check.