Geometry

It is indeed a right triangle, and the shortest altitude will be perpendicular to the hypotenuse. 

The area of the triangle is $10 \times 24 \div 2 = 120$.

Because the altitude is perpendicular to the hypotenuse, we know that $\text{hypotenuse} \times\text {altitude} \div 2 = [\text{ABC}]$. Think of the hypotenuse as the base, and the altitude as the height. 

Subsituting what we know, we have: $26 \times \text{altitude} \div 2 = 120$, and we can solve for the altitude. 

Hope this helps!!!