Step by step
The question
Question: the radius is 25 cm and the chord is 7 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 = 7 and c = 25.
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.
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 make zero.
Cross them out.
625 minus 49 is 576.
6 Take the square root
b² is 576.
So b times b is 576.
That means b is the square root of 576.
The square root of 576 is the positive number that times itself makes 576.
24 times 24 is 576.
So b = 24 cm.
The missing side is 24 cm.
7 Double it
The chord is twice A M: 48 cm.
8 Check
Let's check: put b = 24 cm back.
First a² + b².
Now c².
Both are 625.
So b = 24 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 49.
Then: take the square root.
Then: double it.
Last: check.