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The altitude $ h $ corresponding to a side $ b $ is given by:
A = rac{1}{2} b h \Rightarrow h = rac{2A}{b}
To find the longest altitude, we use the shortest side, since altitude is inversely proportional to the base. The shortest side is $ 7 $ cm.
h_{ ext{longest}} = rac{2 \cdot 20\sqrt{3}}{7} = rac{40\sqrt{3}}{7} ext{ cm}
Thus, the length of the longest altitude is $ oxed{ rac{40\sqrt{3}}{7}} $ cm.Question: A paleobotanist studying fossilized ferns models the growth pattern of a prehistoric plant using the equation $ 4x^2 - 12xy + 9y^2 + 8x - 36y + 20 = 0 $. Determine whether this equation represents a degenerate conic section and, if so, identify the type of degeneracy.
We begin by analyzing the general quadratic form:
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
Given equation:
4x^2 - 12xy + 9y^2 + 8x - 36y + 20 = 0
