Step by step
The question
Question: chords A B and C D cross at P, with AP = 4, PB = 6, CP = 3: find PD.
1 Use the rule
When two chords cross, the two parts of one chord multiply to the same as the two parts of the other.
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.
4 × 6 = 3 × x
Careful!
Multiply the two parts of one chord.
Not a part of one chord with a part of the other.
Careful: part × part of the SAME chord
4 times 6 is 24.
24 = 3 × x
3 Remove the 3
x is times 3.
To remove times 3, we divide both sides by 3: 8.
x = 243 = 8
4 Check
Let's check: both sides give 24.
So it is right!
4 × 6 = 24 3 × 8 = 24
Let's look back.
First: use the rule.
Then: put the numbers in.
Then: remove the 3.
Last: check.
Answer
x = 243 = 8