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i n t r u s i o n - Parker Core Knowledge

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Question: A triangle has side lengths $ 7 $, $ 24 $, and $ 25 $. What is the radius of the inscribed circle?

📌 For a right triangle, the inradius $ r $ is given by:

📌 where $ c $ is the hypotenuse. Substituting:

📌 $$Question: A logistics analyst is modeling delivery routes in a city grid where each block is represented as a unit step east or north. How many different routes starting from the origin reach a point 5 blocks east and 4 blocks north without crossing the line $ y = x + 1 $?

📌 Solution: To count the number of valid lattice paths from $(0, 0)$ to $(5, 4)$ that do not cross the line $ y = x + 1 $, we use the reflection principle. First, compute the total number of unrestricted paths from $(0, 0)$ to $(5, 4)$:

📌 Now, we subtract the number of paths that cross the line $ y = x + 1 $. A path crosses this line if at some point $ y > x + 1 $. To reflect such paths, reflect the starting point across the line $ y = x + 1 $. The reflection of $(0, 0)$ across $ y = x + 1 $ is $(-1, 1)$. The number of reflected paths from $(-1, 1)$ to $(5, 4)$ is:

📌 Thus, the number of valid paths that do not cross $ y = x + 1 $ is:

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