Problem 37 - Algebra 1: Concepts and Skills

Question

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?

Step-by-Step Solution

Answer

They should meet at the point (1,1) and each will have to hike \( \sqrt{13} \) miles to arrive.

Step 1 - Set Up the Coordinates

Set the starting point as the origin (0,0) of the coordinate system. After the hike, the teacher is at the point (-2,3) (2 miles west and 3 miles north) and the friend is at the point (4,-1) (4 miles east and 1 mile south).

Step 2 - Use Midpoint Formula

Calculate the midpoint between two points in the coordinate system by averaging each point's coordinates. The midpoint M, where they should meet, can be found using the formula: \(M = \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right)\). Substituting the points from step 1 results: \(M = \left( \frac{-2 + 4}{2}, \frac{3 + (-1)}{2} \right) = (1,1)\)

Step 3 - Calculate the distance to the midpoint

To calculate the distance they have to hike, use the distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2-y_1)^2}\). For the teacher, the distance is \(d = \sqrt{(1 - (-2))^2 + (1 - 3)^2} = \sqrt{9 + 4} = \sqrt{13}\) miles. To check if the friend will also hike the same distance, calculate the distance for the friend as well: \(d = \sqrt{(1 - 4)^2 + (1 - (-1))^2} = \sqrt{9 + 4} = \sqrt{13}\) miles. Both results match.

Key Concepts

Midpoint Formula

Imagine you're on a treasure hunt and you find out that the treasure is exactly halfway between two landmarks. To find this treasure, you'd need to calculate the midpoint. In the world of algebra, when we're dealing with points on a coordinate plane, we use the midpoint formula for this very purpose.

The midpoint formula helps us find the central point between two distinct points on a graph. It's quite straightforward; just take the average of the x-coordinates and the y-coordinates separately. If you have points A with coordinates \(x_1, y_1\) and B with \(x_2, y_2\), the midpoint, M, is given by:\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
In real-life terms, if you and your friend are hiking from different spots and you want to meet exactly in the middle, you'd use the midpoint formula to find the equally distant spot from both of you. As shown in the exercise solution, the formula helps you ensure that both you and your friend hike the same distance to meet.

Distance Formula

Now that you've planned where to meet your friend using the midpoint formula, you need to figure out how far each of you must hike. This is where the distance formula comes into play. Just like getting the right measurements to build a piece of furniture, we measure 'distances' between points to ensure accuracy.

The distance formula is derived from the Pythagorean theorem and calculates the straight-line distance between two points on the coordinate plane. If we have the coordinates \(x_1, y_1\) and \(x_2, y_2\), the distance d between them is found by:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
In practice, as demonstrated in your hiking scenario, after finding the meeting point, you can determine the exact distance you and your friend need to hike to get to the midpoint. No one hikes more than the other, making your adventure fair and enjoyable.

Coordinate Plane

Picture a map spread out in front of you; it's essentially a coordinate plane—a two-dimensional surface where you can pinpoint every location using two numbers. These numbers are known as coordinates, and the plane is divided by two lines: the horizontal x-axis and the vertical y-axis.

The point where these two axes intersect is called the origin, which is labeled as \(0,0\). Locations on this plane are given as \(x, y\), where 'x' indicates the horizontal distance from the origin, and 'y' signifies the vertical distance. Therefore, a point \(3,2\) means you move 3 units right (east) and 2 units up (north) from the origin.

Back to your hiking scenario, the starting spot was set as the origin. When you moved from this point, every step north, south, east, or west was essentially moving across the coordinate plane. Without such a system, it'd be much harder to plot your exact positions and figure out where to meet. The coordinate plane is a foundational concept in algebra and geometry that allows us to visualize and solve problems related to distances and midpoints.