Step by step
The question
Question: the radius is 10 cm and the chord is 6 cm 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 = 6 and c = 10.
4 Work out the squares
6² means 6 times 6.
That is 36.
10² means 10 times 10.
That is 100.
Write the squares: 36 + b² = 100.
5 Remove the 36
We want b² by itself.
The 36 is in the way.
To remove 36, we take away 36.
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.
36 and − 36 make zero.
Cross them out.
100 minus 36 is 64.
6 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 cm.
The missing side is 8 cm.
7 Double it
The chord is twice A M: 16 cm.
8 Check
Let's check: put b = 8 cm back.
First a² + b².
Now c².
Both are 100.
So b = 8 cm 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 36.
Then: take the square root.
Then: double it.
Last: check.