Answer:
6.5
Explanation:
You want the length of the hypotenuse of right triangle ABC with sin(A) = 12/13 and an inradius of 1.
Right triangle
The ratio of long leg to hypotenuse of 12/13 tells us this is a triangle with sides in the ratios 5 : 12 : 13. For the purpose of determining an inradius, we can assume these are the actual side lengths.
Inradius
The inradius of a triangle is ...

For side lengths 5, 12, 13, we have ...
s = (5+12+13)/2 = 15
r = √((15 -5)(15 -12)(15 -13)/15) = 2
This tells us our triangle with sides 5, 12, 13 is 2 times the size of the one we want.
Hypotenuse
The length of the hypotenuse of ∆ABC is 13/2 = 6.5 units.
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Additional comment
We can use the Pythagorean theorem to find the length of the third side, given a side of 12 and a hypotenuse of 13.
a² +b² = c²
a² +12² = 13²
a = √(169 -144) = √25 = 5
It is easier to consult our memory of Pythagorean triples. The ones most commonly seen in algebra, trig, and geometry problems are ...
{3, 4, 5}, {5, 12, 13}, {7, 24, 25}, {8, 15, 17}, {9, 40, 41}
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