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#### 6Question: An ancient Andean ceremonial platform is shaped like a regular hexagon with side length $ 5 $ meters. If each side is extended outward by $ 2 $ meters to form a larger regular hexagon, by how many square meters does the area increase?
The new side length is $ 5 + 2 = 7 $ meters. The area of the larger hexagon is:
Question: A seismologist models seismic wave propagation in a right triangular fault zone, where the hypotenuse represents the rupture line of length $ r $, and the inradius of the triangle is $ d $. If the triangles legs are in the ratio $ 3:4 $, find the ratio of the area of the incircle to the area of the triangle.
Question: A climatologist analyzes a triangular agricultural plot with side lengths $ 13 $ m, $ 14 $ m, and $ 15 $ m. To optimize irrigation, she needs to install a water line perpendicular to the largest side from the opposite vertex. What is the length of this shortest altitude?
Solution: The shortest altitude corresponds to the longest side, which is $ 15 $ m. First, compute the area using Herons formula. The semiperimeter $ s $ is: