Step by step
The question
Question: find the legs of a 45-45-90 triangle with a hypotenuse of 12.
1 Half of a square
The angles are 45°, 45° and 90°.
This is half of a square.
The longest side cuts the square in two.
So the two legs are sides of the square.
They are equal.
We call both x.
2 Use a² + b² = c²
Both legs are x.
So Pythagoras gives x² + x² = 12².
12² means 12 times 12.
That is 144.
One x² and one more x² make 2x².
So 2x² = 144.
Careful!
The legs are not half of the hypotenuse.
The hypotenuse is a leg times √2, not times 2.
3 Remove the 2
2x² means 2 times x².
To remove times 2, we divide by 2.
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.
2 divided by 2 is 1.
So just x² is left.
144 divided by 2 is 72.
4 Take the square root
x times x is 72.
So x is the square root of 72.
But 72 has a square number in it: 36.
72 is 36 times 2.
A root of a times splits into two roots.
√36 is 6.
So x = 6√2.
That is the exact answer, made simple.
Both legs are 6√2.
5 Check
Let's check: put x = 6√2 in x² + x².
We get 144, the same as 12².
So x = 6√2 is right!
Let's look back.
First: half of a square.
Then: use a² + b² = c².
Then: remove the 2.
Then: take the square root.
Last: check.