The answer is \(\displaystyle 12\).

Since we know that the permieter is \(\displaystyle 40\) inches and one side is \(\displaystyle 12\) inches, it can be determined that the remaining two sides must combine to be \(\displaystyle 40-12=28\) inches. The ratio of the remaining two sides is \(\displaystyle 3:4\) which means 3 parts : 4 parts or 7 parts combined. We can then set up the equation \(\displaystyle 7x=28\), and divide both sides by \(\displaystyle 7\) which means \(\displaystyle x=4\). The ratio of the remaining side lengths then becomes \(\displaystyle 3*(4):4*(4)\) or \(\displaystyle 12:16\). We now know the 3 side lengths are \(\displaystyle 12,12,and\ 16\).

\(\displaystyle 12\) is the shortest side and thus the answer.

To form an isosceles triangle here, we need to create a third vertex whose \(\displaystyle y\) coordinate is between \(\displaystyle -1\) and \(\displaystyle 7\).  If a vertex is placed at \(\displaystyle (2,3)\), the distance from \(\displaystyle (1,-1)\) to this point will be \(\displaystyle \sqrt{17}\). The distance from \(\displaystyle (1,7)\) to this point will be the same.

Because \(\displaystyle \bigtriangleup BDE\) is isosceles, \(\displaystyle \measuredangle DBE\) equals \(\displaystyle \left(\frac{180-58}{2}\right)^{\circ}\) or \(\displaystyle 61^{\circ}\).

We know that \(\displaystyle \measuredangle ABE, \measuredangle DBE, \measuredangle CBD\) add up to \(\displaystyle 180^{\circ}\), so \(\displaystyle \measuredangle CBD\) must equal \(\displaystyle (180-90-61)^{\circ}\) or \(\displaystyle 29^{\circ}\).

This problem can be solved when one realizes that the light beam's split has resulted in an acute isosceles triangle. The triangle as stated has two sides of \(\displaystyle 5\) feet apiece, which meets the requirement for isosceles triangles, and having one angle of \(\displaystyle 30^{\circ}\) at the vertex where the two congruent sides meet means the other two angles must be \(\displaystyle 75^{\circ}\) and \(\displaystyle 75^{\circ}\). The missing side connecting the two sensors, therefore, is opposite the \(\displaystyle 30^{\circ}\) angle.

Since we know at least two angles and at least one side of our triangle, we can use the Law of Sines to calculate the remainder. The Law of Sines says that for any triangle with angles \(\displaystyle A, B\) and \(\displaystyle C,\) and opposite sides \(\displaystyle a, b\) and \(\displaystyle c\):

\(\displaystyle \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\).

Plugging in one of our \(\displaystyle 75^{\circ}\) angles (and its corresponding \(\displaystyle 5\) ft side) into this equation, as well as our \(\displaystyle 30^{\circ}\) angle (and its corresponding unknown side) into this equation gives us:

\(\displaystyle \frac{\sin 75^{\circ}}{5} = \frac{ \sin 30^{\circ}}{x}\)

Next, cross-multiply:

\(\displaystyle \frac{\sin 75^{\circ}}{5} = \frac{\sin 30^{\circ}}{x}\) ---> \(\displaystyle x\sin 75^{\circ} = 5\sin 30^{\circ}\)

Now simplify and solve:

\(\displaystyle x = \frac{5\sin 30^{\circ}}{\sin 75^{\circ}} = 2.588\)

Rounding, we see our missing side is \(\displaystyle 2.65\textup{ feet}\) long.