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**katti****Member**- Registered: 2010-06-15
- Posts: 2

I need help with a tricky puzzle.

Let's say we have a straight line: A---------------------D

The line has two points, B and C. Each of the two points split the line formed by outer points in two with respect to golden section:

A--------B--------C---------D

So AD is to AC as AC is to CD. And AD is to BD as BD is to AB.

The distance between B and C is 8314 cm. How long is the whole line AD?

Any help is greatly appreciated!

Best regards,

Katti

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

Let's say for the moment that AD has a length of 1 and AC has a length of x.

Then CD has a length of (1-x).

Using the matching ratios, we can write the equation 1/x = x/(1-x).

Rearrange to form a quadratic, and this solves to give x = (√5-1)/2.

The diagram is symmetric, in that AC = BD.

Therefore, we can work out distance BC by finding C's distance from the halfway point and doubling it.

The distance from the halfway point will be (√5-2)/2, and naturally doubling that gets us √5-2.

So we now know that 1 is to √5-2 as AD is to BC.

From there, we do 8314/(√5-2) to get the final answer of ~35219cm.

Why did the vector cross the road?

It wanted to be normal.

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**katti****Member**- Registered: 2010-06-15
- Posts: 2

Amazing! What a thorough explanation, thank you!

Best regards,

Katti

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