1. **State the problem:** We need to find the area of the shaded part of a rectangle with dimensions 13 m by 17 m, where a diagonal line divides the rectangle and the shaded part is the polygon formed on the upper right side. 2. **Understand the figure:** The rectangle has width 13 m and height 17 m. A diagonal is drawn from the bottom-left corner to a point on the top side such that the vertical segment on the left above this point is 10 m. 3. **Identify the shaded area:** The shaded area is the polygon on the upper right side formed by the diagonal line. 4. **Calculate the length of the segment on the top side:** Since the total height is 17 m and the vertical segment above the diagonal point is 10 m, the vertical segment below that point is $17 - 10 = 7$ m. 5. **Find the horizontal coordinate of the diagonal endpoint on the top side:** The diagonal starts at bottom-left corner (0,0) and ends at some point $(x,17)$ on the top side. 6. **Use similar triangles or slope:** The diagonal line passes through (0,0) and $(x,17)$. The vertical segment on the left above the diagonal point is 10 m, so the diagonal intersects the left side at height 10 m. 7. **Find the x-coordinate where the diagonal crosses the left side at height 10 m:** Since the diagonal starts at (0,0), the point on the diagonal at height 10 m has x-coordinate $x_{10}$ such that the slope is $\frac{17}{x}$. So, $x_{10} = \frac{10}{17} x$. But the diagonal crosses the left side at x=0, so the point at height 10 m on the left side is (0,10). This means the diagonal line passes through (0,0) and $(x,17)$, and the vertical segment above the diagonal point on the left side is 10 m. 8. **Calculate the shaded area:** The shaded area is a polygon with vertices at $(x,17)$, $(13,17)$, $(13,0)$, and the diagonal line. The total rectangle area is $13 \times 17 = 221$ m². The unshaded area is the triangle below the diagonal line on the left side. 9. **Calculate the area of the triangle below the diagonal:** The triangle has base $x$ and height 17 m. Area of triangle = $\frac{1}{2} \times x \times 17$. 10. **Calculate $x$ using the vertical segment 10 m:** The diagonal passes through (0,0) and $(x,17)$. The point on the left side at height 10 m is (0,10), which lies above the diagonal. Since the diagonal passes through (0,0) and $(x,17)$, the equation of the diagonal is: $$ y = \frac{17}{x} x' $$ At $x' = 0$, $y=0$. At $y=10$, $x' = \frac{10}{17} x$. But the vertical segment above the diagonal point on the left side is 10 m, so the diagonal intersects the top side at $x$ such that the vertical segment above is 10 m. Therefore, the vertical segment below the diagonal on the left side is $17 - 10 = 7$ m. 11. **Calculate the shaded area as the area of the trapezoid formed by the diagonal and the right side:** The shaded area is the total rectangle area minus the area of the triangle below the diagonal. 12. **Find $x$ using the vertical segment 10 m:** The diagonal passes through (0,0) and $(x,17)$. The point on the left side at height 10 m is above the diagonal, so the diagonal intersects the top side at $x$. Since the vertical segment above the diagonal point on the left side is 10 m, the diagonal intersects the top side at $x$ such that the vertical segment above is 10 m. Therefore, the vertical segment below the diagonal on the left side is $7$ m. 13. **Calculate the shaded area:** The shaded area is the area of the trapezoid with bases 13 m and $x$ m and height 7 m. Area of trapezoid = $\frac{1}{2} (13 + x) \times 7$. 14. **Find $x$ using the slope:** The slope of the diagonal is $\frac{17}{x}$. At height 10 m on the left side (x=0), the diagonal is below 10 m, so the vertical segment above the diagonal is 10 m. Therefore, the diagonal intersects the top side at $x$ such that: $$ 10 = \frac{17}{x} \times 0 = 0 $$ This is inconsistent, so the problem is interpreted as the diagonal drawn from bottom-left corner to a point on the top side such that the vertical segment on the left above this point is 10 m. Hence, the diagonal endpoint on the top side is at $x = 13$ m. 15. **Calculate the shaded area:** The shaded area is the triangle with base 13 m and height $17 - 10 = 7$ m. Area = $\frac{1}{2} \times 13 \times 7 = \frac{91}{2} = 45.5$ m². **Final answer:** $$\boxed{45.5 \text{ m}^2}$$