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Home » Since \( f(3) = g(3) \), we have: - Portal da AcÃºstica

Since \( f(3) = g(3) \), we have: - Portal da AcÃºstica

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Pregunta: Un científico de alimentos modela las superficies de proteínas vegetales en treatment cosinesas como triángulos. Para un triángulo rectángulo con hipotenusa $ c $ e inradio $ r $, si $ r = \frac{a + b - c}{2} $, demuestre que el área del triángulo es $ r \cdot s $, donde $ s $ es el semiperímetro.

📌 A) True, by definition

📌 B) True only if acute

📌 C) False, since area is $ r \cdot s $ only if $ r $ is inradius

📌 D) False — area is $ \frac{1}{2}ab $, and $ r s = r \cdot \frac{a+b+c}{2} $, and $ r = \frac{a+b-c}{2} $, so $ r s = \frac{a+b-c}{2} \cdot \frac{a+b+c}{2} = \frac{(a+b)^2 - c^2}{4} = \frac{a^2 + 2ab + b^2 - c^2}{4} $. Since $ c^2 = a^2 + b^2 $, this becomes $ \frac{2ab}{4} = \frac{ab}{2} $ — yes.

📌 E) $ r s = \frac{ab}{2} $ — true

📌 But best choice:

📌 D) False — although $ r s = \frac{ab}{2} $, the statement area equals $ r \cdot s $ is analytically true only when $ r $ is inradius, which it is — so D is misleading.

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