Step by step
The question
Question: the radius is 13 and the chord is 5 from the center: find the length of the chord AB.
1 Use the chord rule
The line from the center at 90° to a chord cuts the chord in half.
M is the middle of A B.
So O, M and A make a right triangle.
The right angle is at M.
O A goes from the center to the circle: it is a radius.
2 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.
3 Put the numbers in
Now put the numbers in: a = 5 and c = 13.
4 Work out the squares
5² means 5 times 5.
That is 25.
13² means 13 times 13.
That is 169.
Write the squares: 25 + b² = 169.
5 Remove the 25
We want b² by itself.
The 25 is in the way.
To remove 25, we take away 25.
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.
25 and − 25 make zero.
Cross them out.
169 minus 25 is 144.
6 Take the square root
b² is 144.
So b times b is 144.
That means b is the square root of 144.
The square root of 144 is the positive number that times itself makes 144.
12 times 12 is 144.
So b = 12.
The missing side is 12.
7 Double it
The chord is twice A M: 24.
8 Check
Let's check: put b = 12 back.
First a² + b².
Now c².
Both are 169.
So b = 12 is right!
Let's look back.
First: use the chord rule.
Then: name the sides.
Then: put the numbers in.
Then: work out the squares.
Then: remove the 25.
Then: take the square root.
Then: double it.
Last: check.