thank you for the info: but how do you make sure that BC is smaller than AC given that the picture might not be drawn to scale?

how do you get the part 1+\frac{BC}{AC} ?
thank you, please help

Good question,

But you can try putting the some values,

But remember it should satisfy the equation \(\frac{AB}{AC}=\frac{AC}{BC}\)

Let AB =12
AC= 6
BC = 3

These values satisfy the above equation , and \(\frac{AC}{BC}= 2\) which is less than 3

Because AB = AC + BC

Now
\(\frac{AB}{AC}\) can be written as

\(\frac{(AC +BC)}{AC} = \frac{AC}{AC} + \frac{BC}{AC}\)

= \(1 + \frac{BC}{AC}\)

I think if AB=12 and AC = 6 then BC must equal to 6 and not 3 as BC = AB-AC

BC must be smaller, otherwise the equation in the question cannot hold.
Try to plug in numbers (with BC > AC) and you'll see

In case you're familiar with the golden ratio, that's what this question is. Short length over medium length = medium length over whole. The golden ratio is ~1.6

Okay here since 1+ BC/AC = AC/BC. for AC/BC>1 from previous equation AC>BC and from that we can say BC/AC <1 and 1+(<1)<2

therefore AC/BC<2

and therefore quantity B is greater than quantity A

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