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.
=
=
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
⇒
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.