Step by step
The question
Question: find the missing side of a right triangle with leg 3 and longest side 6.
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 = 3 and c = 6.
3 Work out the squares
3² means 3 times 3.
That is 9.
6² means 6 times 6.
That is 36.
Write the squares: 9 + b² = 36.
4 Remove the 9
We want b² by itself.
The 9 is in the way.
To remove 9, we take away 9.
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.
9 and − 9 make zero.
Cross them out.
36 minus 9 is 27.
5 Take the square root
b² is 27.
So b times b is 27.
That means b is the square root of 27.
27 is not a square number.
It is between 5 and 6.
But 27 has a square number in it: 9.
27 is 9 times 3.
A root of a times splits into two roots.
√9 is 3.
So b = 3√3.
That is the exact answer, made simple.
With a calculator, 3√3 is about 5.2, to the nearest tenth.
So b is about 5.2.
6 Check
Let's check: put b = 3√3 back.
First a² + b².
(3√3)² is 3² times (√3)²: 9 times 3.
Now c².
Both are 36.
So b = 3√3 is right!
Let's look back.
First: name the sides.
Then: put the numbers in.
Then: work out the squares.
Then: remove the 9.
Then: take the square root.
Last: check.