Answer:
24 ft
Explanation:
so, we don't know anything else about the triangle ?
ok, let's see.
the area of a triangle is (a side length) × (the height from that side to the opposite corner) / 2
At = 24 = side × height / 2
48 = side × height = 10 × height
height = 48/10 = 4.8 = 24/5
let's say that the height on our known side splits this side into 2 parts, p and q (p+q = 10).
we can calculate the triangle side on the right hand side of our know side by calling it a and using Pythagoras :
a² = height² + q² = (4.8)² + q² = 23.04 + q²
as all sides have to be even integers, a² has to be an even square number larger than 23.04.
and because p+q = 10, we know q must be smaller than 10, and therefore q² smaller than 100.
the only candidates for a² are therefore 36 and 64 (6² and 8²).
in a similar way this applies to the left hand triangle side b tool.
b² = height² + p² = 23.04 + p²
with the same restrictions and possible solutions as a².
we have the possibilities that a = b = 6 or 8, or a = 6 and b = 8 (or vice versa).
let's rule out a=b :
a=b wound also mean p=q=5
then
a² = 23.04 + 5² = 23.04 + 25 = 48.04, which is not an even square integer. therefore, this assumption is wrong.
so, the only possible solution is a = 6 and b = 8 (or vice versa, but it did not matter which is which, as we only need the perimeter, which would be the same either way).
proof :
36 = 23.04 + 12.96 = 23.04 + q²
=> q = 3.6 ft
64 = 23.04 + 40.96 = 23.04 + p²
=> p = 6.4 ft
p+q = 3.6 + 6.4 = 10 ft
perfect, it fits, this is the correct solution
so, the perimeter of the triangle is
10 + 6 + 8 = 24 ft