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Home » Breakthrough Alert: Half-Life 2’s Long-Awaited Release Date Finally Revealed! - RTA

Breakthrough Alert: Half-Life 2’s Long-Awaited Release Date Finally Revealed! - RTA

📅 May 10, 2026 👤 scraface

May 10, 2026

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📌 Question: A circle is circumscribed around an equilateral triangle with side length $ s $. If the radius of the circle is $ R $, express $ R $ in terms of $ s $.

📌 Solution: In an equilateral triangle, the circumradius $ R $ is related to the side length $ s $ by the formula:

📌 But this is incorrect; the correct formula comes from geometry. In an equilateral triangle, the centroid, circumcenter, and orthocenter coincide. The height $ h $ of the triangle is:

📌 The circumradius is two-thirds of the height (from vertex to centroid):

📌 The new area is increased by 50%, so:

📌 $$Question: A glaciologist analyzing satellite images observes 7 distinct ice fractures in a section of a glacier. If she selects 4 fractures at random to model their connectivity using numerical simulations, what is the probability that a specific critical fracture is included in the selection?

📌 Solution: We are given a total of 7 distinct ice fractures, and the glaciologist selects 4 of them at random. We are to compute the probability that a specific critical fracture (say, Fracture A) is among the 4 selected.

📌 The total number of ways to choose 4 fractures out of 7 is given by the binomial coefficient:

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