How can the hypotenuse alone determine the scaling factor here?

Module 2 Week 3 Day 13 Challenge Part 1 Explanation
How can the hypotenuse alone determine the scaling factor here?

@thebladedancer Hi there! This is an interesting question. There is a long train of logic that precedes this fact.
Why is the ratio of the triangle areas based entirely on the ratio of the triangles' bases?
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It's because their heights are the same. ← Why is this true?
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\(\hspace{ 40 mm} \) It's because the two right triangles are congruent.
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\(\hspace{20 mm} \) Why are the two triangles congruent?\(\hspace{40 mm} \) ↑
\(\hspace{20 mm}\) It's because they each have a right angle and the \(\textcolor{green}{\text{green}}\) angle.\(\hspace{40 mm} \) ↑
\(\hspace{20 mm}\) Why do they both have a \(\textcolor{green}{\text{green}}\) angle?
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\(\hspace{20 mm} \) It's because of Angle Bisector Theorem!
This is how Prof. Loh gets a ratio of \(\frac{(1+\sqrt{5})^2}{2^2}\) for the area of the larger inner triangle to the area of the smaller inner triangle. 
po shn loh crazy hand 2.0