Step by step
The question
Question: P T is a tangent and P A B a secant, with PA = 2, PB = 8: find PT.
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 PT x.
x2 = 2 × 8
Careful!
The tangent is on both sides of its triangle pair, so it is squared.
Careful: PT is squared: PT × PT
2 times 8 is 16.
x2 = 16
3 Take the square root
x squared is 16, so x is the square root.
A length is positive: 4.
x = √16 = 4
4 Check
Let's check: both sides give 16.
So it is right!
4² = 16 2 × 8 = 16
Let's look back.
First: use the rule.
Then: put the numbers in.
Then: take the square root.
Last: check.
Answer
x = √16 = 4