Step by step
The question
Question: find PT.
PT is a tangent to the circle with center O.
The radius is 6 cm.
OP = 10 cm.
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 = 6 and c = 10.
4 Work out the squares
6² means 6 times 6.
That is 36.
10² means 10 times 10.
That is 100.
Write the squares: 36 + b² = 100.
5 Remove the 36
We want b² by itself.
The 36 is in the way.
To remove 36, we take away 36.
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.
36 and − 36 make zero.
Cross them out.
100 minus 36 is 64.
6 Take the square root
b² is 64.
So b times b is 64.
That means b is the square root of 64.
The square root of 64 is the positive number that times itself makes 64.
8 times 8 is 64.
So b = 8 cm.
The missing side is 8 cm.
7 Check
Let's check: put b = 8 cm back.
First a² + b².
Now c².
Both are 100.
So b = 8 cm 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 36.
Then: take the square root.
Last: check.