Inside the $7\times8$ rectangle below, one point is chosen a distance $\sqrt2$ from the left side and a distance $\sqrt7$ from the bottom side. The line segments from that point to the four vertices of the rectangle are drawn. Find the area of the shaded region.
[asy]
import graph;
size(4cm);
pair A = (0,0);
pair B = (9,0);
pair C = (9,7);
pair D = (0,7);
pair P = (1.5,3);
draw(A--B--C--D--cycle,linewidth(1.5));
filldraw(A--B--P--cycle,rgb(.76,.76,.76),linewidth(1.5));
filldraw(C--D--P--cycle,rgb(.76,.76,.76),linewidth(1.5));
[/asy]
1. **Identify the coordinates of the point inside the rectangle:**
   - The point is chosen a distance $\sqrt{2}$ from the left side and a distance $\sqrt{7}$ from the bottom side.
   - Therefore, the coordinates of the point are $(\sqrt{2}, \sqrt{7})$.

2. **Determine the vertices of the rectangle:**
   - The rectangle is $7 \times 8$, so the vertices are:
     - Bottom-left: $(0, 0)$
     - Bottom-right: $(8, 0)$
     - Top-right: $(8, 7)$
     - Top-left: $(0, 7)$

3. **Draw the line segments from the point to the vertices:**
   - The point $(\sqrt{2}, \sqrt{7})$ is connected to the four vertices of the rectangle.

4. **Divide the rectangle into four triangles:**
   - The point $(\sqrt{2}, \sqrt{7})$ divides the rectangle into four triangles:
     - Triangle 1: with vertices $(0, 0)$, $(8, 0)$, and $(\sqrt{2}, \sqrt{7})$
     - Triangle 2: with vertices $(8, 0)$, $(8, 7)$, and $(\sqrt{2}, \sqrt{7})$
     - Triangle 3: with vertices $(8, 7)$, $(0, 7)$, and $(\sqrt{2}, \sqrt{7})$
     - Triangle 4: with vertices $(0, 7)$, $(0, 0)$, and $(\sqrt{2}, \sqrt{7})$

5. **Calculate the area of each triangle:**
   - The area of a triangle is given by the formula:
     \[
     \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
     \]
   - For Triangle 1:
     \[
     \text{Area}_1 = \frac{1}{2} \left| 0(0 - \sqrt{7}) + 8(\sqrt{7} - 0) + \sqrt{2}(0 - 0) \right| = \frac{1}{2} \left| 0 + 8\sqrt{7} + 0 \right| = 4\sqrt{7}
     \]
   - For Triangle 2:
     \[
     \text{Area}_2 = \frac{1}{2} \left| 8(7 - \sqrt{7}) + 8(\sqrt{7} - \sqrt{7}) + \sqrt{2}(\sqrt{7} - 7) \right| = \frac{1}{2} \left| 8(7 - \sqrt{7}) + 0 + \sqrt{2}(\sqrt{7} - 7) \right|
     \]
     Simplifying:
     \[
     \text{Area}_2 = \frac{1}{2} \left| 56 - 8\sqrt{7} + \sqrt{2}(\sqrt{7} - 7) \right|
     \]
   - For Triangle 3:
     \[
     \text{Area}_3 = \frac{1}{2} \left| 8(7 - \sqrt{7}) + 0 + \sqrt{2}(\sqrt{7} - 7) \right|
     \]
   - For Triangle 4:
     \[
     \text{Area}_4 = \frac{1}{2} \left| 0(7 - 0) + 0(0 - \sqrt{7}) + \sqrt{2}(0 - 7) \right| = \frac{1}{2} \left| 0 + 0 - 7\sqrt{2} \right| = \frac{7\sqrt{2}}{2}
     \]

6. **Sum the areas of the triangles:**
   - The total area of the shaded region is the sum of the areas of the triangles:
     \[
     \text{Total Area} = \text{Area}_1 + \text{Area}_2 + \text{Area}_3 + \text{Area}_4
     \]

7. **Simplify the expression:**
   - Since the combined height of both triangles is the height of the rectangle, their combined area is:
     \[
     \text{Total Area} = \frac{7 \times 8}{2} = 28
     \]

The final answer is $\boxed{28}$