MHB Geocaching: Solve a+b=c c/b=b/a Problem

[TheCricketer]

I love geocaching and a problem that recently came up is:

A--------------B------------C (not to scale)

Find the distance of AB if the proportion of AC:BC is equal to BC:AB. Substituting a for AB, b for BC and c for AC, I come up with:

a+b=c where c/b=b/a

I'm at a loss here. When I put it in WA it comes up with the answer (although looking at the answer, I have no idea how it calculated it) and I was able to "brute force" my answer as well and get the right answer (thanks Excel) but I want to know the math behind it. The actual values are irrelevant as I want to get to the simplified formulas.

 

[HOI]

Surely, you realize that you can't determine the actual distance from A to B knowing only proportions?

What, exactly, did you put into WA?

 

[tkhunny]

You have relationships. Without actual values, that's all you have.

c = a+b -- good

c/b = b/a -- good

You could also write $\dfrac{a+b}{b} = \dfrac{b}{a}$

You could also write $a^{2}+ab = b^{2}$

You could also write $(a-b)^{2} = ab$

You could also write: $a = \dfrac{b}{2}(\sqrt{5}-1)$

Which simplified formulas would you like?

 

[TheCricketer]

You have relationships. Without actual values, that's all you have.

c = a+b -- good

c/b = b/a -- good

You could also write $\dfrac{a+b}{b} = \dfrac{b}{a}$

You could also write $a^{2}+ab = b^{2}$

You could also write $(a-b)^{2} = ab$

You could also write: $a = \dfrac{b}{2}(\sqrt{5}-1)$

Which simplified formulas would you like?

Can you simplify for b please and show the work? I don’t know where the $\sqrt{5}$ comes from.

 

[tkhunny]

Can you simplify for b please and show the work? I don’t know where the $\sqrt{5}$ comes from.

This is where you get to demonstrate your love of mathematics. I used the Quadratic Formula with the expression immediately above. There are a few other ways to do it. Show your work!