Step by step
The question
Question: find PT.
PT is a tangent to the circle with center O.
The radius is 5.
OP = 13.
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 PT.
3 Put the numbers in
Now put the numbers in: a = 5 and c = 13.
4 Work out the squares
5² means 5 times 5.
That is 25.
13² means 13 times 13.
That is 169.
Write the squares: 25 + b² = 169.
5 Remove the 25
We want b² by itself.
The 25 is in the way.
To remove 25, we take away 25.
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.
25 and − 25 make zero.
Cross them out.
169 minus 25 is 144.
6 Take the square root
b² is 144.
So b times b is 144.
That means b is the square root of 144.
The square root of 144 is the positive number that times itself makes 144.
12 times 12 is 144.
So b = 12.
The missing side is 12.
7 Check
Let's check: put b = 12 back.
First a² + b².
Now c².
Both are 169.
So b = 12 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 25.
Then: take the square root.
Last: check.