Question

1. complete the square box problem.

3
5

Explanation:

Step1: Recall property of square diagonals

The diagonals of a square bisect each other and are perpendicular. The four - triangles formed by the diagonals of a square are congruent right - isosceles triangles.

Step2: Analyze the given numbers

Let's assume the two non - shaded adjacent right - isosceles triangles have areas 3 and 5. Let the side lengths of the right - isosceles triangles be \(a\) and \(b\) respectively. The area of a right - isosceles triangle is \(A=\frac{1}{2}x^{2}\), where \(x\) is the length of the equal sides. If \(\frac{1}{2}a^{2}=3\) and \(\frac{1}{2}b^{2}=5\).

Step3: Find the area of the shaded region

The shaded region is also a right - isosceles triangle. Notice that if we consider the relationships between the areas of the right - isosceles triangles formed by the diagonals of the square. The area of the shaded triangle \(A_s\) satisfies the following relationship based on the fact that the triangles are similar. The area of the shaded triangle is \(\sqrt{3\times5}=\sqrt{15}\).

Answer:

\(\sqrt{15}\)