The two shortest sides of a right triangle are 10 in. and 24 in. long. what is the length of the shortest side of a similar right triangle whose two longest sides are 36 in. and 39 in?

If the two shortest sides of the triangle are 10in and 24in, then using Pythagoras' theorem, the longest side =
 
= 
= 

Now we know the two longest sides of the first triangle (24in and 26in) we can compare them with the two longest sides of the second triangle.
If  = the scale factor the first triangle is enlarged by then
and  
⇒ 

Finally, we need to multiply the smallest side of the first triangle by the scale factor to find the shortest side of the second triangle.

So the length of the shortest side of the other triangle is 15in.

You could, instead, calculate the length of the shortest side of the second triangle by using Pythagoras' theorem and ignoring the first triangle completely.