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how to watch osu game saturday - Anadea

📅 May 16, 2026 👤 scraface

May 16, 2026

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📌 A right triangle has legs of length 9 and 12. A circle is inscribed in the triangle. What is the radius of the circle?

📌 The area \( A = \frac{1}{2} \cdot 9 \cdot 12 = 54 \).

📌 The semi-perimeter \( s = \frac{9 + 12 + 15}{2} = 18 \), where 15 is the hypotenuse (via Pythagoras).

📌 The inradius \( r = \frac{A}{s} = \frac{54}{18} = 3 \).

📌 The function \( f(x) = 2^{x} + 2^{-x} \) has a minimum value. What is this minimum value?

📌 Let \( y = 2^x \), so \( f(x) = y + \frac{1}{y} \).

📌 By AM-GM, \( y + \frac{1}{y} \geq 2 \), with equality when \( y = 1 \), i.e., \( x = 0 \).

📌 Thus, the minimum value is 2.

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