Step by step
The question
Question: find the missing side of a right triangle with leg 7 and longest side 25.
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
25 is the longest side, so it is c.
a is 7, so we write 7 in place of a.
c is 25, so we write 25 in place of c.
b stays b: it is the side we want.
4 Work out the squares
7² means 7 times 7.
That is 49.
25² means 25 times 25.
That is 625.
Write the squares: 49 + b² = 625.
Is b alone now?
Not yet: + 49 is still with b.
How do we remove it?
5 Remove the 49
We want b² by itself.
The 49 is in the way.
To remove 49, we take away 49.
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.
49 and − 49 cancel each other.
They make zero.
625 minus 49 is 576.
So the square on side b is 576.
So after step 5: we took away 49, and now we have b² = 576.
6 Take the square root
Now we have b² = 576.
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 576.
The square root of 576 is the plus number that times itself makes 576.
24 times 24 is 576.
−24 times −24 is 576 too, but the √ sign always means the plus one.
So b = 24.
The missing side is 24.
7 Check
Let's check: put b = 24 back.
First a² + b².
Now c².
Both are 625.
So b = 24 is right!
8 Pick the choice
Our answer is b = 24.
That is choice B.
9 Why not the others
A is 18.
Wrong: 25 − 7: took the sides away, not their squares.
C is √674.
Wrong: added the squares: for a leg we take away.
D is 576.
Wrong: 625 − 49 = 576, 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: remove the 49.
Then: take the square root.
Then: check.
Then: pick the choice.
Last: why not the others.