Worked examples
The question
Question: the radius is 5 and the chord is 8 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: 8 divided by 2 is 4.
AM = 8 ÷ 2 = 4
Careful!
The right triangle has only half of the chord.
A M is half of A B.
Careful: not AM = 8 ✗
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.
a2 + b2 = c2
4 Put the numbers in
Now put the numbers in: a = 4 and c = 5.
42 + b2 = 52
5 Work out the squares
4² means 4 times 4.
That is 16.
42 = 4 × 4 = 16
5² means 5 times 5.
That is 25.
52 = 5 × 5 = 25
Write the squares: 16 + b² = 25.
16 + b2 = 25
6 Remove the 16
We want b² by itself.
The 16 is in the way.
To remove 16, we take away 16.
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.
16 + b2 − 16 = 25 − 16
16 and − 16 make zero.
Cross them out.
25 minus 16 is 9.
b2 = 9
7 Take the square root
b² is 9.
So b times b is 9.
That means b is the square root of 9.
b = √9
The square root of 9 is the positive number that times itself makes 9.
3 times 3 is 9.
3 × 3 = 9
So b = 3.
b = 3
The missing side is 3.
8 Check
Let's check: put b = 3 back.
First a² + b².
a2 + b2 = 42 + 32 = 16 + 9 = 25
Now c².
c2 = 52 = 25
Both are 25.
So b = 3 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 16.
Then: take the square root.
Last: check.
▶ Watch it solved
The question
Question: the chord is 24 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: 24 divided by 2 is 12 cm.
AM = 24 ÷ 2 = 12 cm
Careful!
The right triangle has only half of the chord.
A M is half of A B.
Careful: not AM = 24 ✗
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.
a2 + b2 = c2
4 Put the numbers in
Now put the numbers in: a = 12 and b = 5.
122 + 52 = c2
5 Work out the squares
12² means 12 times 12.
That is 144.
122 = 12 × 12 = 144
5² means 5 times 5.
That is 25.
52 = 5 × 5 = 25
Write the squares: 144 + 25 = c².
144 + 25 = c2
144 plus 25 is 169.
169 = c2
6 Take the square root
c² is 169.
So c times c is 169.
That means c is the square root of 169.
c = √169
The square root of 169 is the positive number that times itself makes 169.
13 times 13 is 169.
13 × 13 = 169
So c = 13 cm.
c = 13 cm
The missing side is 13 cm.
7 Check
Let's check: put c = 13 cm back.
First a² + b².
a2 + b2 = 122 + 52 = 144 + 25 = 169
Now c².
c2 = 132 = 169
Both are 169.
So c = 13 cm 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.
▶ Watch it solved
The question
Question: the radius is 6 and the chord is 8 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: 8 divided by 2 is 4.
AM = 8 ÷ 2 = 4
Careful!
The right triangle has only half of the chord.
A M is half of A B.
Careful: not AM = 8 ✗
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.
a2 + b2 = c2
4 Put the numbers in
Now put the numbers in: a = 4 and c = 6.
42 + b2 = 62
5 Work out the squares
4² means 4 times 4.
That is 16.
42 = 4 × 4 = 16
6² means 6 times 6.
That is 36.
62 = 6 × 6 = 36
Write the squares: 16 + b² = 36.
16 + b2 = 36
6 Remove the 16
We want b² by itself.
The 16 is in the way.
To remove 16, we take away 16.
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.
16 + b2 − 16 = 36 − 16
16 and − 16 make zero.
Cross them out.
36 minus 16 is 20.
b2 = 20
7 Take the square root
b² is 20.
So b times b is 20.
That means b is the square root of 20.
b = √20
20 is not a square number.
It is between 4 and 5.
4 × 4 = 16, 5 × 5 = 25
But 20 has a square number in it: 4.
20 is 4 times 5.
b = √4 × 5
A root of a times splits into two roots.
b = √4 × √5
√4 is 2.
2 × 2 = 4
So b = 2√5.
That is the exact answer, made simple.
b = 2√5
With a calculator, 2√5 is about 4.5, to the nearest tenth.
So b is about 4.5.
b ≈ 4.5
8 Check
Let's check: put b = 2√5 back.
First a² + b².
(2√5)² is 2² times (√5)²: 4 times 5.
a2 + b2 = 42 + (2√5)2 = 16 + 4 × 5
= 16 + 20 = 36
Now c².
c2 = 62 = 36
Both are 36.
So b = 2√5 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 16.
Then: take the square root.
Last: check.
▶ Watch it solved
