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Unit 1 Geometry Help

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KidwaiMaaz001:
Hi,
So I've figured out how to prove that AD/BD = BD/CD, but I don't get how to prove the second part of the question; that the aforementioned ratios are equal to φ - 1. How do I prove this?
Thanks.

fun_jirachi:
Hey there! :)

If we have the radius of the circle being 1, and the side length of the square being \(s\), we can ascertain that \(s = \sqrt{1-\frac{s^2}{4}}\)  ie. we have that \(\frac{5s^2}{4} = 1 \implies s = \frac{2}{\sqrt{5}}\).
Hence, we have that \(AD = 1-\frac{1}{\sqrt{5}}\) and \(CD = 1+\frac{1}{\sqrt{5}}\), with \(BD = \frac{2}{\sqrt{5}}\). What we also note here is \(\phi - 1 = \frac{\sqrt{5}-1}{2}\) - we can also see that \(\frac{AD}{BD} = \frac{1-\frac{1}{\sqrt{5}}}{\frac{2}{\sqrt{5}}} = \frac{\sqrt{5}-1}{2}\), and this holds true also for \(\frac{BD}{CD}\). Since \(\phi\) is a constant, we can just verify the ratios with the value given in the question - this holds true for any mathematical constant you might come across!

Hope this helps :)

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