Worked examples
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.
▶ Watch it solved
The question
Question: two secants P A B and P C D come from P, with PA = 4, PB = 10, PC = 5: 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.
4 × 10 = 5 × 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
4 times 10 is 40.
40 = 5 × x
3 Remove the 5
x is times 5.
To remove times 5, we divide both sides by 5: 8.
x = 405 = 8
4 Check
Let's check: both sides give 40.
So it is right!
4 × 10 = 40 5 × 8 = 40
Let's look back.
First: use the rule.
Then: put the numbers in.
Then: remove the 5.
Last: check.
▶ Watch it solved
The question
Question: P T is a tangent and P A B a secant, with PA = 4, PB = 9: 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 = 4 × 9
Careful!
The tangent is on both sides of its triangle pair, so it is squared.
Careful: PT is squared: PT × PT
4 times 9 is 36.
x2 = 36
3 Take the square root
x squared is 36, so x is the square root.
A length is positive: 6.
x = √36 = 6
4 Check
Let's check: both sides give 36.
So it is right!
6² = 36 4 × 9 = 36
Let's look back.
First: use the rule.
Then: put the numbers in.
Then: take the square root.
Last: check.
▶ Watch it solved
