1. **Problem Statement:** We have a square with diagonals intersecting at right angles, forming a smaller tilted square inside. We need to find the approximate value of $x$, which is marked on the diagonal near the smaller square. 2. **Key Properties:** In a square, the diagonals are equal in length and bisect each other at right angles. The diagonals form four right triangles inside the square. 3. **Diagonal Length:** Let the side length of the square be $s$. The diagonal length $d$ is given by the Pythagorean theorem: $$d = s\sqrt{2}$$ 4. **Smaller Square:** The smaller square formed by the diagonals has sides equal to half the diagonal of the larger square because the diagonals intersect at their midpoints. 5. **Finding $x$:** The value $x$ is the length from the center to the vertex of the smaller square along the diagonal. This length is half the diagonal of the smaller square. 6. **Calculate $x$:** Since the smaller square's side is $\frac{d}{\sqrt{2}} = \frac{s\sqrt{2}}{\sqrt{2}} = s$, the diagonal of the smaller square is: $$d_{small} = s\sqrt{2}$$ Half of this diagonal is: $$x = \frac{d_{small}}{2} = \frac{s\sqrt{2}}{2} = s \times 0.707$$ 7. **Approximate $x$:** If the side length $s$ is approximately 4.0 units (from the options and typical square size), then: $$x \approx 4.0 \times 0.707 = 2.828$$ 8. **Conclusion:** The approximate value of $x$ is 2.8 units. **Final answer:** 2.8 units