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. If you and your friend want to meet for lunch, where could you meet so that both of you hike the same distance? How far do you have to hike?

Short answer

You and your friend have to hike 3.85 miles to meet. You have to move an additional 0.25 miles towards the east or southeast, and your friend has to move back 0.25 miles, in the westerly or northwesterly direction to reach the meeting point

Step-by-step solution

Calculate individual traveled distance

First, find the distance each person has hiked. The trip is split into two parts, so compute it using Pythagoras theorem. For us, the distance is \( \sqrt{(3^2 + 2^2)} = 3.6 \) miles. For the friend, the distance is \( \sqrt{(4^2 + 1^2)} = 4.1 \) miles.

Calculate the meeting point

We both should walk the same distance to meet. Hence, to meet halfway, we should both travel halfway of the total distance that we both have. The sum of the distances is \( 3.6 + 4.1 = 7.7 \) miles. Half of this distance is \( \frac{7.7}{2} = 3.85 \) miles

Find out how far we have to move

We've already traveled 3.6 miles and to meet halfway, we need to travel 3.85 miles. Thus, to reach the meeting point, we need to move an additional \( 3.85 - 3.6 = 0.25 \) miles.

Find out how far our friend has to move

Our friend has already traveled 4.1 miles. To get to the meeting point, which is 3.85 miles from the starting point, our friend has to move back by \( 4.1 - 3.85 = 0.25 \) miles.

Key concepts

Distance formula

Understanding the distance formula is essential for solving many types of geometric problems, including the one presented in our exercise. The distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) is derived from the Pythagorean theorem and helps to find the distance between any two points in a coordinate plane.

When applied to our hiking problem, we treated each hiker's path as a leg of a right triangle, with one leg representing the north-south direction, and the other the east-west direction. By using the distances traveled as the lengths of these legs, we can calculate the direct path — the hypotenuse — they each traversed.

For real-world applications, remember that the coordinates \(x_1, y_1\) and \(x_2, y_2\) correspond to the starting and ending points. This formula is essential not only for mathematicians but also for various professionals like architects, engineers, and even sports strategists who need to calculate exact distances.

Geometric Distance

Geometric distance refers to the shortest path between two points in space. It's crucial in various fields, such as navigation, computer graphics, and robotics. In our hiking example, we are interested in finding the shortest path that each person would need to travel to reach a mutual meeting point.

Since the paths taken by the hikers form right-angled triangles, we implemented the geometric principle that the shortest distance between two points is a straight line — in this case, the hypotenuse. The geometric distance gives us a precise measurement of how far each person is from the meeting point, regardless of the route they took.

This concept is not limited to two dimensions; it can be extended to three-dimensional space, which is incredibly useful for air and sea navigation where elevation must be considered in addition to latitude and longitude.

Problem-solving in Algebra

Problem-solving in algebra involves using algebraic methods to solve equations and word problems. It requires an understanding of how to apply theorems and formulas in practical situations. In the case of our hiking scenario, algebraic problem-solving allowed us to calculate distances and determine where the two hikers should meet.

To solve the problem, we defined variables, created equations based on the Pythagorean theorem, and solved them step by step. This methodical approach is critical in algebra, as it helps break down complex problems into manageable parts. With this approach, we could find the additional distance each hiker needs to walk to meet each other.

By practicing problem-solving in algebra, students enhance their analytical skills and ability to approach a wide range of challenges that may not only be limited to mathematical exercises but can also include real-life situations that require logical thinking and precision.