1. **Problem statement:** Calculate the length of the segment $PM$ in the given 3D prism with a trapezoidal base, where $M$ is the midpoint of rectangle $ABCD$. 2. **Understanding the figure and given data:** - The prism has a trapezoidal base with points $A, B, C, D$. - $M$ is the midpoint of rectangle $ABCD$. - Given lengths: $AB = 6$ cm, $BC = 7$ cm, depth $= 8$ cm, height $= 17$ cm. 3. **Step 1: Find coordinates for points to use the 3D distance formula.** - Place $A$ at origin $(0,0,0)$. - Since $AB = 6$ cm along x-axis, $B = (6,0,0)$. - $BC = 7$ cm vertically, so $C = (6,7,0)$. - $D$ completes the rectangle, so $D = (0,7,0)$. - $M$ is midpoint of $ABCD$, so $M = \left(\frac{0+6+6+0}{4}, \frac{0+0+7+7}{4}, 0\right) = (3,3.5,0)$. 4. **Step 2: Point $P$ is at height 17 cm above $A$ (assuming vertical height).** - So $P = (0,0,17)$. 5. **Step 3: Use the 3D distance formula to find $PM$:** $$PM = \sqrt{(x_P - x_M)^2 + (y_P - y_M)^2 + (z_P - z_M)^2}$$ $$= \sqrt{(0 - 3)^2 + (0 - 3.5)^2 + (17 - 0)^2}$$ $$= \sqrt{(-3)^2 + (-3.5)^2 + 17^2}$$ $$= \sqrt{9 + 12.25 + 289}$$ $$= \sqrt{310.25}$$ $$= 17.61 \text{ cm (approx)}$$ **Final answer:** The length of segment $PM$ is approximately $17.61$ cm.