Step by step
The question
Question: find PT.
PT is a tangent to the circle with center O.
The radius is 7.
OP = 9.
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 = 7 and c = 9.
4 Work out the squares
7² means 7 times 7.
That is 49.
9² means 9 times 9.
That is 81.
Write the squares: 49 + b² = 81.
5 Remove the 49
We want b² by itself.
The 49 is in the way.
To remove 49, we take away 49.
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.
49 and − 49 make zero.
Cross them out.
81 minus 49 is 32.
6 Take the square root
b² is 32.
So b times b is 32.
That means b is the square root of 32.
32 is not a square number.
It is between 5 and 6.
But 32 has a square number in it: 16.
32 is 16 times 2.
A root of a times splits into two roots.
√16 is 4.
So b = 4√2.
That is the exact answer, made simple.
With a calculator, 4√2 is about 5.7, to the nearest tenth.
So b is about 5.7.
7 Check
Let's check: put b = 4√2 back.
First a² + b².
(4√2)² is 4² times (√2)²: 16 times 2.
Now c².
Both are 81.
So b = 4√2 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 49.
Then: take the square root.
Last: check.