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Home » The inradius $ r $ of a right triangle is given by the formula: - Ready Digital AB

The inradius $ r $ of a right triangle is given by the formula: - Ready Digital AB

📅 March 15, 2026 👤 scraface

Mar 15, 2026

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📌 A = I + \frac{B}{2} - 1

📌 where $A$ is area, $I$ is interior lattice points, and $B$ is boundary lattice points. We seek $I$, but the problem wants interior points only. First, compute the area of the large equilateral triangle with side length $6$:

📌 A = \frac{\sqrt{3}}{4} \cdot 6^2 = \frac{\sqrt{3}}{4} \cdot 36 = 9\sqrt{3}

📌 Now, determine $B$: the number of lattice points on the boundary. Each side has $6 + 1 = 7$ points, but corners are shared. Since it’s a triangle,

📌 B = 3 \cdot 7 - 3 = 21 - 3 = 18

📌 Now apply Pick’s Theorem:

📌 9\sqrt{3} = I + \frac{18}{2} - 1 = I + 9 - 1 = I + 8

📌 I = 9\sqrt{3} - 8

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