Problem 54 - Algebra 1

Question

Use the following information. You and a friend go hiking. You hike 3 miles north and 2 miles west. Starting from the same point, your friend hikes 4 miles east and 1 mile south. How far apart are you and your friend? (Hint: Draw a diagram on a grid.)

Step-by-Step Solution

Answer

Based on the solution steps, the distance between you and your friend is approximately 4.47 miles.

Step 1: Represent the Paths on a Graph

Visualising problems can often make them easier to understand. Start by representing yours and your friend's path on a Cartesian plane or grid. Let's say the starting point is the origin (0,0). Your pathway can be represented by the vector (2,3) and your friend's by (4,-1) which means, You end up at the point (2,3) and your friend at the point (4,-1).

Step 2: Calculate the Difference in x and y Coordinates

The difference in the x coordinates between your and your friend's final positions is: \(dx = x_2 - x_1 = 4 - 2 = 2\). The difference in the y coordinates between your and your friend's final positions is: \(dy = y_2 - y_1 = -1 - 3 = -4\). These values dx (horizontal distance) and dy (vertical distance) represent the sides of a right triangle where the distance we want to calculate is the hypotenuse.

Step 3: Apply Pythagorean Theorem

The Pythagorean theorem states that for any right triangle with sides of lengths a, b (dx and dy in our case), and c (hypotenuse), the relationship between these lengths is: \(a^2 + b^2 = c^2\). Substituting the dx and dy values, we get: \(c^2 = 2^2 + (-4)^2 = 4 + 16 = 20\). Hence, \(c = \sqrt{20} \approx 4.47\) miles.

Key Concepts

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that helps us find the length of the hypotenuse in a right-angled triangle. The theorem states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Written as an equation, it looks like this: \( a^2 + b^2 = c^2 \). Here, \( a \) and \( b \) are the lengths of the triangle's legs and \( c \) is the hypotenuse. Applying this theorem is crucial when solving distance-related problems in a coordinate system.

• Step-by-step usage: Identify the two shorter sides of your right triangle (often represented by line segments on a graph).
• Calculate the square of each side, sum the squares, and finally find the square root of this sum to determine the hypotenuse.

The Pythagorean theorem is particularly useful in real-world scenarios, such as determining distances on a map or even navigating when hiking in an unknown area, just like in our original exercise.

Coordinate Graphing

Coordinate graphing provides us with a visual way to solve problems involving positions and distances. A coordinate plane consists of two perpendicular axes, labeled the x-axis (horizontal) and y-axis (vertical). Any point on this plane is represented by a pair of numbers known as coordinates, \( (x, y) \). In our hiking scenario, the starting point \( (0, 0) \) was given, which is the origin of the graph.

• How to plot points: To plot a point, move along the x-axis to the x-coordinate and then move parallel to the y-axis to the y-coordinate.
• Understanding movement: When you hike north or south, you change your y-coordinate; going north increases it, while going south decreases it. East and west movements change the x-coordinate; east increases it, while west decreases it.

By plotting these movements on a coordinate graph, represented as vectors, we can easily determine the ending positions before calculating the distance between the two hikers.

Vectors

Vectors are a powerful mathematical construct used to represent quantities that have both magnitude and direction. A vector is typically represented in a coordinate plane as an arrow pointing from one point to another. In the context of hiking, vectors help illustrate the change in position on the graph.

• Representation: A vector \( \vec{v} \) from point \( (0,0) \) to \( (x,y) \) is written as \( \langle x, y \rangle \).
• Operations: Vectors can be added together to find a resultant vector, showing overall movement, or subtracted to find the difference, such as in our original exercise where we subtract coordinates to find \( dx \) and \( dy \).

In the hiking problem, vectors \( (2, 3) \) for you and \( (4, -1) \) for your friend represent the paths on a graph. These vectors help visualize the problem and find the relative position and distance between the two hikers, making complex spatial problems more understandable with simple arithmetic and visualization.