Question

1. the hshs backpacking club decided to split up. half of the hiking group hiked 4 miles east and 2 miles north. the other half of the group hiked 3 miles west and 4 miles south. how far apart are the two groups? graph it

Explanation:

Step1: Determine the coordinates

Assume the starting - point is \((0,0)\).
For the first group:
If they hike \(4\) miles east (positive \(x\) - direction) and \(2\) miles north (positive \(y\) - direction), their coordinates are \((x_1,y_1)=(4,2)\).
For the second group:
If they hike \(3\) miles west (negative \(x\) - direction) and \(4\) miles south (negative \(y\) - direction), their coordinates are \((x_2,y_2)=(- 3,-4)\).

Step2: Use the distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
Substitute \(x_1 = 4,y_1 = 2,x_2=-3,y_2 = - 4\) into the formula:
\(x_2 - x_1=-3 - 4=-7\) and \(y_2 - y_1=-4 - 2=-6\).
Then \(d=\sqrt{(-7)^2+(-6)^2}=\sqrt{49 + 36}\).

Step3: Calculate the distance

\(d=\sqrt{85}\approx9.22\) miles.

Answer:

The two groups are approximately \(\sqrt{85}\approx9.22\) miles apart.