Step by step
The question
Question: find OP.
PT is a tangent to the circle with center O.
The radius is 5.
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!
The longest side is always across from the right angle.
Here that is OP.
3 Put the numbers in
Now put the numbers in: a = 5 and b = 12.
4 Work out the squares
5² means 5 times 5.
That is 25.
12² means 12 times 12.
That is 144.
Write the squares: 25 + 144 = c².
25 plus 144 is 169.
5 Take the square root
c² is 169.
So c times c is 169.
That means c is the square root of 169.
The square root of 169 is the positive number that times itself makes 169.
13 times 13 is 169.
So c = 13.
The missing side is 13.
6 Check
Let's check: put c = 13 back.
First a² + b².
Now c².
Both are 169.
So c = 13 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: take the square root.
Last: check.