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5Question: A paleobotanist is analyzing a triangular fossil fragment with side lengths of 7 cm, 8 cm, and 9 cm. What is the length of the shortest altitude drawn to the longest side?
To find the shortest altitude, we note that the shortest altitude is drawn to the longest side (9 cm).
First, we compute the area of the triangle using Herons formula.
Let $ a = 7 $, $ b = 8 $, $ c = 9 $.
The altitude $ h $ to the side of length 9 cm is given by:
Question: A climatologist models a triangular region on a map with side lengths of 5 km, 6 km, and 7 km. What is the radius of the circumscribed circle around this triangle?
We use the formula for the circumradius $ R $ of a triangle:
where $ a = 5 $, $ b = 6 $, $ c = 7 $, and $ A $ is the area.
Question: A neural interface developer models a triangular electrode pad with sides of 10 cm, 10 cm, and 12 cm. What is the radius of the inscribed circle?
This is an isosceles triangle with sides $ a = 10 $, $ b = 10 $, $ c = 12 $.
