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📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 5Question: A coastal geographer uses remote sensing to study a triangular marine reserve with side lengths of 13 km, 14 km, and 15 km. What is the length of the shortest altitude, in kilometers, drawn to the longest side?

📌 Solution: We are given a triangle with sides $ a = 13 $, $ b = 14 $, and $ c = 15 $. The longest side is $ c = 15 $, so we are to find the shortest altitude to this side—this corresponds to the altitude from the vertex opposite the 15 km side.

📌 First, compute the area of the triangle using Herons formula. The semi-perimeter is:

📌 Thus, the altitude to the longest side is $ 11.2 $ km. Since this is the altitude to the longest side, and the other altitudes to shorter sides would be longer (as area is fixed), this is the shortest altitude.

📌 Question: An electrical engineer designs a high-voltage transmission tower base shaped like a regular hexagon with a side length of 8 meters. What is the radius of the circumcircle surrounding the hexagon, in meters?

📌 Solution: A regular hexagon can be divided into 6 equilateral triangles, each with side length equal to the radius of the circumcircle. Since the side length of the hexagon is 8 meters, the distance from the center to any vertex (the circumradius) is also 8 meters.

📌 This is a well-known geometric property: in a regular hexagon, the side length equals the circumradius.

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