Step by step
The question
Question: find the missing side of a right triangle with leg 8 and longest side 17.
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, and it is true for every right triangle.
3 Put the numbers in
17 is the longest side, so it is c.
a is 8, so we write 8 in place of a.
c is 17, so we write 17 in place of c.
b stays b: it is the side we want.
4 Work out the squares
8² means 8 times 8.
That is 64.
17² means 17 times 17.
That is 289.
Write the squares: 64 + b² = 289.
Is b alone now?
Not yet: + 64 is still with b.
How do we remove it?
5 Remove the 64
We want b² by itself.
The 64 is in the way.
To remove 64, we take away 64.
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.
64 and − 64 cancel each other.
They make zero.
289 minus 64 is 225.
So the square on side b is 225.
So after step 5: we took away 64, and now we have b² = 225.
6 Take the square root
Now we have b² = 225.
b² means b times b.
But we want b alone, so we need to remove the ².
To remove a square, we take the square root: b is the square root of 225.
The square root of 225 is the plus number that times itself makes 225.
15 times 15 is 225.
−15 times −15 is 225 too, but the √ sign always means the plus one.
So b = 15.
The missing side is 15.
7 Check
Let's check: put b = 15 back.
First a² + b².
Now c².
Both are 289.
So b = 15 is right!
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 64.
Then: take the square root.
Last: check.