Step by step
The question
Question: P T is a tangent and P A B a secant, with PT = 10, PA = 5: find PB.
1 Use the rule
The tangent squared is the outside part of the secant times the whole secant.
Why?
The two small triangles have the same angles, so they are similar.
Their sides go in the same ratio.
2 Put the numbers in
Put the numbers in.
We call PB x.
102 = 5 × x
Careful!
The tangent is on both sides of its triangle pair, so it is squared.
Careful: PT is squared: PT × PT
10 times 10 is 100.
100 = 5 × x
3 Remove the 5
x is times 5.
To remove times 5, we divide both sides by 5: 20.
x = 1005 = 20
4 Check
Let's check: both sides give 100.
So it is right!
10² = 100 5 × 20 = 100
Let's look back.
First: use the rule.
Then: put the numbers in.
Then: remove the 5.
Last: check.
Answer
x = 1005 = 20