Step by step
The question
Question: find the longest side of a right triangle with legs 5 and 5.
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 = 5 and b = 5.
3 Work out the squares
5² means 5 times 5.
That is 25.
5² means 5 times 5.
That is 25.
Write the squares: 25 + 25 = c².
25 plus 25 is 50.
4 Take the square root
c² is 50.
So c times c is 50.
That means c is the square root of 50.
50 is not a square number.
It is between 7 and 8.
But 50 has a square number in it: 25.
50 is 25 times 2.
A root of a times splits into two roots.
√25 is 5.
So c = 5√2.
That is the exact answer, made simple.
With a calculator, 5√2 is about 7.1, to the nearest tenth.
So c is about 7.1.
5 Check
Let's check: put c = 5√2 back.
First a² + b².
Now c².
(5√2)² is 5² times (√2)²: 25 times 2.
Both are 50.
So c = 5√2 is right!
Let's look back.
First: name the sides.
Then: put the numbers in.
Then: work out the squares.
Then: take the square root.
Last: check.