Three ballet dancers are positioned on stage. David is 7 feet straight behind Megan and 4 feet directly left of Tom. When the music begins, David twirls to Tom's position, then leaps to Megan's position, and finally walks back to his original position. How far did David travel? If necessary, round to the nearest tenth. \( \square \) feet

To find the distance David traveled, we need to calculate the distances he moved from his starting position to Tom, then to Megan, and back to his starting position. 1. **David's position relative to Megan and Tom:** - Let’s place Megan at the origin (0, 0). - Consequently, Megan's position is at (0, 0). - David is 7 feet behind Megan, so his position is (0, -7). - David is also 4 feet left of Tom, which means if Tom's position is (x, y), then David's position would be (x - 4, y). 2. **Finding Tom's coordinates:** - Since David is at (0, -7) and is 4 feet left of Tom, we have: \( x - 4 = 0 \) (which gives us \( x = 4 \)) - David is directly left of Tom, so Tom is at (4, -7). 3. **Calculating distances:** - **Distance from David to Tom:** Using the distance formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) \( d_{D\rightarrow T} = \sqrt{(4 - 0)^2 + (-7 + 7)^2} = \sqrt{(4)^2 + (0)^2} = 4 \) feet. - **Distance from Tom to Megan:** \( d_{T\rightarrow M} = \sqrt{(0 - 4)^2 + (0 + 7)^2} = \sqrt{(-4)^2 + (7)^2} = \sqrt{16 + 49} = \sqrt{65} \) feet, approximately 8.1 feet. - **Distance from Megan back to David:** \( d_{M\rightarrow D} = \sqrt{(0 - 0)^2 + (0 + 7)^2} = \sqrt{(0)^2 + (7)^2} = 7 \) feet. 4. **Total distance traveled by David:** \( \text{Total Distance} = d_{D\rightarrow T} + d_{T\rightarrow M} + d_{M\rightarrow D} = 4 + 8.1 + 7 \approx 19.1 \) feet. Thus, David traveled approximately **19.1 feet**.