Step by step
The question
Question: find the longest side of a right triangle with legs 8 and 15.
There are four choices.
First we solve it, the normal way.
Then we pick.
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
a is 8, so we write 8 in place of a.
b is 15, so we write 15 in place of b.
c stays c: it is the side we want.
4 Work out the squares
8² means 8 times 8.
That is 64.
15² means 15 times 15.
That is 225.
Write the squares: 64 + 225 = c².
64 plus 225 is 289.
The two small squares make 289.
So the big square is 289 too.
5 Take the square root
Now we have 289 = c².
c² means c times c.
But we want c alone, so we need to remove the ².
To remove a square, we take the square root: c is the square root of 289.
The square root of 289 is the plus number that times itself makes 289.
17 times 17 is 289.
−17 times −17 is 289 too, but the √ sign always means the plus one.
So c = 17.
The missing side is 17.
6 Check
Let's check: put c = 17 back.
First a² + b².
Now c².
Both are 289.
So c = 17 is right!
7 Pick the choice
Our answer is c = 17.
That is choice B.
8 Why not the others
A is 7.
Wrong: 15 − 8: took the legs away.
C is 23.
Wrong: 8 + 15: added the legs.
D is 289.
Wrong: 64 + 225 = 289, but we still had to take the square root.
Let's look back.
First: name the sides.
Then: squares on the sides.
Then: put the numbers in.
Then: work out the squares.
Then: take the square root.
Then: check.
Then: pick the choice.
Last: why not the others.