A hiker goes 3 miles north, 2 miles east, one mile south, 4 miles east and then 2 miles north. After an injury, he sends his coordinates to a helicopter rescue team. If the helicopter starts at the same place as the hiker did, and it flew in a straight path to the hikers ending location, approximately how many miles did the helicopter have to fly? Solve using pythsgorean theorem and diagram
northern direction
= 3 - 1 + 2 = 4
eastern direction
= 2 + 4 = 6
h^2 = 4^2 + 6^2 = 52
the hypotenuse is √52 = appr. 7.2 miles
To solve this problem using the Pythagorean theorem and a diagram, let's first visualize the hiker's journey:
Starting at the origin point, the hiker goes:
- 3 miles north (upward on the y-axis)
- 2 miles east (to the right on the x-axis)
- 1 mile south (downward on the y-axis)
- 4 miles east (to the right on the x-axis)
- 2 miles north (upward on the y-axis)
Drawing a diagram, we can label the coordinates:
Hiker's starting point: (0, 0)
Coordinates after each move:
1. (0, 3)
2. (2, 3)
3. (2, 2)
4. (6, 2)
5. (6, 4)
Now, let's find the distance between the hiker's starting point and the final coordinates (6, 4) using the Pythagorean theorem:
Using the Pythagorean theorem: c^2 = a^2 + b^2
where c is the hypotenuse, and a and b are the other two sides.
The horizontal distance (a) is 6 units, and the vertical distance (b) is 4 units. Hence, we have:
c^2 = 6^2 + 4^2
c^2 = 36 + 16
c^2 = 52
Taking the square root of both sides:
c ≈ √52
c ≈ 7.21
Therefore, the approximate distance the helicopter had to fly to reach the hiker's final location is 7.21 miles.
To solve this problem using the Pythagorean theorem, we need to first visualize the hiker's path and then calculate the distance between the starting point and the ending location.
Let's represent the starting point as (0, 0) on a Cartesian coordinate plane. The hiker's path can be represented as a series of vectors: 3 miles north, 2 miles east, 1 mile south, 4 miles east, and 2 miles north.
Using this representation, the hiker's ending location can be calculated by summing the vector components:
(0, 0) + (0, 3) + (2, 3) - (2, 4) + (0, 2) = (0, 4)
So, the hiker's ending location is (0, 4).
Next, we calculate the distance between the starting point (0, 0) and the ending location (0, 4) using the Pythagorean theorem. The distance formula is given by:
Distance = √((x2 - x1)² + (y2 - y1)²)
In this case, x1 = 0, y1 = 0, x2 = 0, and y2 = 4. Plugging these values into the formula, we get:
Distance = √((0 - 0)² + (4 - 0)²)
= √(0 + 16)
= √16
= 4
Therefore, the helicopter had to fly approximately 4 miles to reach the hiker's ending location.