find the exact length of the missing side of the right triangle

Explanation:

Step1: Apply Pythagorean theorem

In a right - triangle, (a^{2}+b^{2}=c^{2}), where (c) is the hypotenuse. Here, if the hypotenuse (c = \sqrt{88}) and one side (a=\sqrt{56}), and the other side is (y). Then (y^{2}+(\sqrt{56})^{2}=(\sqrt{88})^{2}).

Step2: Simplify the equation

We know that ((\sqrt{m})^{2}=m) for (m\geq0). So the equation becomes (y^{2}+56 = 88).

Step3: Solve for (y^{2})

Subtract 56 from both sides of the equation: (y^{2}=88 - 56), so (y^{2}=32).

Step4: Solve for (y)

Take the square - root of both sides. Since (y) represents the length of a side of a triangle, (y>0). So (y=\sqrt{32}=\sqrt{16\times2}=4\sqrt{2}).

Answer:

(4\sqrt{2})