Step by step
The question
Question: find the radius OT.
PT is a tangent to the circle with center O.
OP = 13.
PT = 12.
1 Use the tangent rule
PT touches the circle at T, and nowhere else.
It is called a tangent.
OT goes from the center to the touch point: a radius.
The radius meets the tangent at a right angle.
This is always true for a tangent.
So O, T and P make a right triangle.
The right angle is at T.
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.
Careful!
OP is across from the right angle.
So OP is the longest side, c.
Not OT.
3 Put the numbers in
Now put the numbers in: a = 12 and c = 13.
4 Work out the squares
12² means 12 times 12.
That is 144.
13² means 13 times 13.
That is 169.
Write the squares: 144 + b² = 169.
5 Remove the 144
We want b² by itself.
The 144 is in the way.
To remove 144, we take away 144.
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.
144 and − 144 make zero.
Cross them out.
169 minus 144 is 25.
6 Take the square root
b² is 25.
So b times b is 25.
That means b is the square root of 25.
The square root of 25 is the positive number that times itself makes 25.
5 times 5 is 25.
So b = 5.
The missing side is 5.
7 Check
Let's check: put b = 5 back.
First a² + b².
Now c².
Both are 169.
So b = 5 is right!
Let's look back.
First: use the tangent rule.
Then: name the sides.
Then: put the numbers in.
Then: work out the squares.
Then: remove the 144.
Then: take the square root.
Last: check.