Step by step
The question
Question: find PT.
PT is a tangent to the circle with center O.
The radius is 6.
OP = 10.
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.
Why?
T is the point of the tangent closest to O.
Every other point of it is outside the circle, so further away.
The shortest way to a line meets it at a right angle.
So O, T and P make a right triangle.
The right angle is at T.
OP is across from it: the longest side.
2 Name the sides
The longest side is called the hypotenuse.
We call it c.
It is across from the right angle, the biggest angle.
The two shorter sides make the right angle.
We call them a and b.
Either one can be a.
3 Squares on the sides
The rule for every right triangle: a² + b² = c².
Where does it come from?
Careful!
OP is across from the right angle.
So OP is the longest side, c.
Not PT.
Draw a square on each side.
a² means a times a.
That is the area of the square on side a.
b² is the area of the square on side b.
c² is the area of the big square, on the longest side.
In every right triangle, the two small squares together make the big square.
This was proved long ago, and it is true for every right triangle.
4 Put the numbers in
10 is the longest side, so it is c.
a is 6, so we write 6 in place of a.
c is 10, so we write 10 in place of c.
b stays b: it is the side we want.
5 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.
Is b alone now?
Not yet: + 36 is still with b.
How do we remove it?
6 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 cancel each other.
They make zero.
100 minus 36 is 64.
So the square on side b is 64.
So after step 6: we took away 36, and now we have b² = 64.
7 Take the square root
Now we have b² = 64.
b² means b times b.
But we want b alone, so we need to remove the ².
To remove a square, we take the square root: b is the square root of 64.
The square root of 64 is the plus number that times itself makes 64.
8 times 8 is 64.
−8 times −8 is 64 too, but the √ sign always means the plus one.
So b = 8.
The missing side is 8.
b is P T.
So P T is 8.
8 Check
Let's check: put b = 8 back.
First a² + b².
Now c².
Both are 100.
So b = 8 is right!
Let's look back.
First: use the tangent rule.
Then: name the sides.
Then: squares on the sides.
Then: put the numbers in.
Then: work out the squares.
Then: remove the 36.
Then: take the square root.
Last: check.