Step by step
The question
Question: the chord is 12 cm long and the chord is 5 cm from the center: find the radius.
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: 12 divided by 2 is 6 cm.
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 = 6 and b = 5.
5 Work out the squares
6² means 6 times 6.
That is 36.
5² means 5 times 5.
That is 25.
Write the squares: 36 + 25 = c².
36 plus 25 is 61.
6 Take the square root
c² is 61.
So c times c is 61.
That means c is the square root of 61.
61 is not a square number.
So we keep √61.
That is the exact answer.
It is between 7 and 8.
With a calculator, √61 is about 7.8, to the nearest tenth.
So c is about 7.8 cm.
7 Check
Let's check: put c = √61 back.
First a² + b².
Now c².
√61 times √61 is 61.
Both are 61.
So c = √61 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: take the square root.
Last: check.