Question

Find integers j and k such that​ (a) the two given integers and j represent the side lengths of an acute triangle and​ (b) the two given integers and k represent the side lengths of an obtuse triangle. 44​, 8

Answer 1

Consider the obtuse triangle. Its long side will be longer than the hypotenuse of an isosceles right triangle with side lengths k. Thus
j > k√2

The triangle inequality requires that the sum of the short legs be longer than the long leg, hence
2k > j

These inequaltities put bounds on the ratio of j to k:
2 > j/k > √2

Some pairs of small integers that satisfy this requirement are
(j, k) ∈ {(3, 2), (5, 3), (6,4), (7, 4), (8, 5), ...}

_____
The first attachment shows the triangles for j=3, k=2.
The second attachment shows the restrictions on the allowable integers and plots the points above.