Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫  π  -¹ ² ³ °

You are not logged in.

## #1 2010-06-15 23:48:16

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

### Can you help me solve a puzzle about the golden section?

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

Offline

## #2 2010-06-16 00:30:08

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

### Re: Can you help me solve a puzzle about the golden section?

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.

Offline

## #3 2010-06-16 05:33:05

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

### Re: Can you help me solve a puzzle about the golden section?

Amazing! What a thorough explanation, thank you!

Best regards,
Katti

Offline