How many units are in the sum of the lengths of the two longest altitudes in a triangle with sides 8, 12, and 17?
Calculate the heights of the triangle from its area.
There are many ways to find the height of the triangle. The easiest way is from the area and base length. The triangle area is half of the product of the base's length and height. Every side of the triangle can be a base; there are three bases and three heights (altitudes). Triangle height is the perpendicular line segment from a vertex to a line containing the base.
Obtuse scalene triangle.
Sides: a = 12 b = 8 c = 17
Area: T = 43.519
Perimeter: p = 37
Semiperimeter: s = 18.5
Angle ∠ A = α = 39.791° = 39°47'28″ = 0.694 rad
Angle ∠ B = β = 25.256° = 25°15'21″ = 0.441 rad
Angle ∠ C = γ = 114.953° = 114°57'11″ = 2.006 rad
Height: h(a) = 7.253
Height: h(b) = 10.88
Height: h(c) = 5.12
7.253 + 10.880 ==18.133 - sum of the 2 longest altitudes of the triangle.
Here is my doodle
Heron's formula, and a calculator for it can be found at https://www.mathsisfun.com/geometry/herons-formula.html