By Heron's formula, the area of the triangle is 90. So the shortest altitude, which is opposite the longest side, is 2*90/26 = 90/13.
+678 Heron's formula is \(\sqrt{s(s-a)(s-b)(s-c)}=area\) where s is the semi-perimeter (in less fancy words \(\frac{area}{2}\)) and a, b, c are the sides of the triangle. I'm pretty sure the guest did something wrong in the calculations there and that is why he/she was wrong
Right scalene triangle.
Sides: a = 10 b = 24 c = 26
Area: T = 120
Perimeter: p = 60
Semiperimeter: s = 30
Angle ∠ A = α = 22.62° = 22°37'11″ = 0.395 rad
Angle ∠ B = β = 67.38° = 67°22'49″ = 1.176 rad
Angle ∠ C = γ = 90° = 1.571 rad
Height: ha = 24
Height: hb = 10
Height: hc = 9.231 ==[2 x 120] / 26 ==9.231 - The shortest altitude.
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