Step by step
The question
Question: find OP.
PT is a tangent to the circle with center O.
The radius is 4.
PT = 6.
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 = 4 and b = 6.
4 Work out the squares
4² means 4 times 4.
That is 16.
6² means 6 times 6.
That is 36.
Write the squares: 16 + 36 = c².
16 plus 36 is 52.
5 Take the square root
c² is 52.
So c times c is 52.
That means c is the square root of 52.
52 is not a square number.
It is between 7 and 8.
But 52 has a square number in it: 4.
52 is 4 times 13.
A root of a times splits into two roots.
√4 is 2.
So c = 2√13.
That is the exact answer, made simple.
With a calculator, 2√13 is about 7.2, to the nearest tenth.
So c is about 7.2.
6 Check
Let's check: put c = 2√13 back.
First a² + b².
Now c².
(2√13)² is 2² times (√13)²: 4 times 13.
Both are 52.
So c = 2√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.