Step by step
The question
Question: find x.
PA and PB are tangents from P to the circle with center O.
PA = 2x + 3 and PB = 11.
1 Use the tangent rule
PA and PB touch the circle at A and at B.
Both are tangents from the point P.
Join the center O to A and to B.
OA and OB are both radii, so they are equal.
Each radius meets its tangent at a right angle.
OP is a side of both triangles.
Right angle, hypotenuse OP, side: the two triangles are congruent, by R H S.
So PA = PB.
Two tangents from one point are always equal: 2x + 3 = 11.
Careful!
The two tangents are the same length.
So one equals the other.
We want x by itself.
Two things are stuck to x: times 2 and + 3.
The + 3 is on the outside, so it comes off first.
Then the times 2.
2 Remove the + 3
To remove + 3, we take away 3.
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.
+ 3 and − 3 make zero.
Cross them out.
11 − 3 = 8.
Now we have 2x = 8.
3 Remove the 2
2x means 2 times x.
We want just x.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
So just x is left.
8 divided by 2 is 4.
So x = 4!
4 Check
Let's check: put x = 4 in each tangent.
We get 11, the same as the other tangent.
So x = 4 is right!
Let's look back.
First: use the tangent rule.
Then: remove the + 3.
Then: remove the 2.
Last: check.