1. **Stating the problem:** We have a cube with side length 10 cm. Point M is the midpoint of segment TS. A string runs from point P to point Q passing through M. We need to find the length of the string PM + MQ. 2. **Understanding the cube and points:** - The cube has edges of length 10 cm. - M is the midpoint of TS, so $TM = MS = \frac{10}{2} = 5$ cm. 3. **Coordinates assignment:** Let's assign coordinates to points for easier calculation. - Let T be at $(0,0,0)$. - Since the cube has side 10, S is at $(10,0,0)$. - M is midpoint of TS, so $M = \left(\frac{0+10}{2}, \frac{0+0}{2}, \frac{0+0}{2}\right) = (5,0,0)$. Assuming P and Q are vertices of the cube, let's assign: - P at $(0,10,0)$ (top front left corner) - Q at $(10,10,10)$ (top back right corner) 4. **Calculate length PM:** $$PM = \sqrt{(5-0)^2 + (0-10)^2 + (0-0)^2} = \sqrt{5^2 + (-10)^2 + 0} = \sqrt{25 + 100} = \sqrt{125} = 5\sqrt{5}$$ 5. **Calculate length MQ:** $$MQ = \sqrt{(10-5)^2 + (10-0)^2 + (10-0)^2} = \sqrt{5^2 + 10^2 + 10^2} = \sqrt{25 + 100 + 100} = \sqrt{225} = 15$$ 6. **Total length of string:** $$PM + MQ = 5\sqrt{5} + 15 \approx 5 \times 2.236 + 15 = 11.18 + 15 = 26.18$$ cm **Final answer:** The length of the string is approximately $26.18$ cm.