Step by step
The question
Question: the radius is 17 and the chord is 30 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: 30 divided by 2 is 15.
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 = 15 and c = 17.
5 Work out the squares
15² means 15 times 15.
That is 225.
17² means 17 times 17.
That is 289.
Write the squares: 225 + b² = 289.
6 Remove the 225
We want b² by itself.
The 225 is in the way.
To remove 225, we take away 225.
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.
225 and − 225 make zero.
Cross them out.
289 minus 225 is 64.
7 Take the square root
b² is 64.
So b times b is 64.
That means b is the square root of 64.
The square root of 64 is the positive number that times itself makes 64.
8 times 8 is 64.
So b = 8.
The missing side is 8.
8 Check
Let's check: put b = 8 back.
First a² + b².
Now c².
Both are 289.
So b = 8 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 225.
Then: take the square root.
Last: check.