Step by step
The question
Question: two secants P A B and P C D come from P, with PA = 3, PB = 12, PC = 4: find PD.
1 Use the rule
For two secants from one point: the outside part times the whole line is the same for both.
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 PD x.
3 × 12 = 4 × x
Careful!
P B is the whole secant, not only the part inside the circle.
Careful: PB is the whole line, from P to B
3 times 12 is 36.
36 = 4 × x
3 Remove the 4
x is times 4.
To remove times 4, we divide both sides by 4: 9.
x = 364 = 9
4 Check
Let's check: both sides give 36.
So it is right!
3 × 12 = 36 4 × 9 = 36
Let's look back.
First: use the rule.
Then: put the numbers in.
Then: remove the 4.
Last: check.
Answer
x = 364 = 9