Step by step
The question
Question: find x.
PT is a tangent to the circle with center O.
The angle at O is 2x.
The angle at P is x.
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: 90°.
This is always true for a tangent.
2 Use the angle fact
O, T and P make a triangle.
Its three angles add up to 180°.
So 2x + x + 90 = 180.
Careful!
The right angle at T is one of the three angles.
It goes in the sum too.
3 Group like terms
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
All the x terms go together.
2x + x = 3x.
Now we have 3x + 90 = 180.
We want x by itself.
Two things are stuck to x: times 3 and + 90.
The + 90 is on the outside, so it comes off first.
Then the times 3.
4 Remove the + 90
To remove + 90, we take away 90.
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.
+ 90 and − 90 make zero.
Cross them out.
180 − 90 = 90.
Now we have 3x = 90.
5 Remove the 3
3x means 3 times x.
We want just x.
To remove times 3, we divide by 3.
We do it on both sides, so they stay equal.
3 divided by 3 is 1.
So just x is left.
90 divided by 3 is 30.
So x = 30!
6 Check
Let's check: put x = 30 in each angle.
Now add the three angles.
They add up to 180°.
So x = 30 is right!
Let's look back.
First: use the tangent rule.
Then: use the angle fact.
Then: group like terms.
Then: remove the + 90.
Then: remove the 3.
Last: check.