Step by step
The question
Question: the radius is 15 and the chord is 18 long: find the distance OM from the center to the chord.
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 Half the chord
A M is half of the chord: 18 divided by 2 is 9.
Careful!
The right triangle has only half of the chord.
A M is half of A B.
3 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.
4 Put the numbers in
Now put the numbers in: a = 9 and c = 15.
5 Work out the squares
9² means 9 times 9.
That is 81.
15² means 15 times 15.
That is 225.
Write the squares: 81 + b² = 225.
6 Remove the 81
We want b² by itself.
The 81 is in the way.
To remove 81, we take away 81.
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.
81 and − 81 make zero.
Cross them out.
225 minus 81 is 144.
7 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.
8 Check
Let's check: put b = 12 back.
First a² + b².
Now c².
Both are 225.
So b = 12 is right!
Let's look back.
First: use the chord rule.
Then: half the chord.
Then: name the sides.
Then: put the numbers in.
Then: work out the squares.
Then: remove the 81.
Then: take the square root.
Last: check.