How can the hypotenuse alone determine the scaling factor here?

]]>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?

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\(\hspace{20 mm}\) It's because they each have a right angle and the \(\textcolor{green}{\text{green}}\) angle.

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\(\hspace{20 mm}\) 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.

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