Step by step
The question
Question: find x.
PA and PB are tangents from P to the circle with center O.
PA = 5x − 6 and PB = 2x + 9.
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: 5x − 6 = 2x + 9.
Careful!
The two tangents are the same length.
So one equals the other.
2 Get x on one side
There is x on both sides.
We want x on one side only.
So we remove the 2x from the right side.
To remove 2x, we take away 2x.
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.
2x and − 2x make zero.
Cross them out.
5x − 2x = 3x.
Now we have 3x − 6 = 9.
We want x by itself.
Two things are stuck to x: times 3 and − 6.
The − 6 is on the outside, so it comes off first.
Then the times 3.
3 Remove the − 6
To remove − 6, we add 6.
We do it on both sides, so they stay equal.
− 6 and + 6 make zero.
Cross them out.
9 + 6 = 15.
Now we have 3x = 15.
4 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.
15 divided by 3 is 5.
So x = 5!
5 Check
Let's check: put x = 5 in each tangent.
Both are 19.
So x = 5 is right!
Let's look back.
First: use the tangent rule.
Then: get x on one side.
Then: remove the − 6.
Then: remove the 3.
Last: check.