Shahriyar is at a winter park with his family they go sledding first then ice skating then skiing what is the total distance traveled between these location ice skating (2, -4) or skiing (-4, -8) sledding (2, 12) to miles

1 answer

To calculate the total distance traveled between the three locations (sledding, ice skating, and skiing), we need to find the distances between each pair of points using the distance formula:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Given the coordinates:

  • Sledding: (2, 12)
  • Ice Skating: (2, -4)
  • Skiing: (-4, -8)

Step 1: Calculate the distance from Sledding to Ice Skating

Let \( (x_1, y_1) = (2, 12) \) and \( (x_2, y_2) = (2, -4) \).

\[ d_{\text{sledding to ice skating}} = \sqrt{(2 - 2)^2 + (-4 - 12)^2} \] \[ = \sqrt{0 + (-16)^2} \] \[ = \sqrt{256} = 16 \]

Step 2: Calculate the distance from Ice Skating to Skiing

Let \( (x_1, y_1) = (2, -4) \) and \( (x_2, y_2) = (-4, -8) \).

\[ d_{\text{ice skating to skiing}} = \sqrt{(-4 - 2)^2 + (-8 + 4)^2} \] \[ = \sqrt{(-6)^2 + (-4)^2} \] \[ = \sqrt{36 + 16} \] \[ = \sqrt{52} = 2\sqrt{13} \approx 7.21 \]

Step 3: Calculate the total distance traveled

Now, sum the two distances:

\[ \text{Total Distance} = 16 + 2\sqrt{13} \]

Approximate Value

\[ 2\sqrt{13} \approx 7.21 \] So,

\[ \text{Total Distance} \approx 16 + 7.21 \approx 23.21 \]

Conclusion

The total distance traveled is approximately 23.21 miles.