Billy’s hometown is mapped on a coordinate grid with the origin being at City Hall. Billy’s house is located at the point (8, 7) and his best friend’s house is located at (2, -1). Where is the midpoint between the two houses located?

Answer

Midpoint Between Two Points on a Coordinate Grid – Complete Guide

How to Find the Midpoint Between Two Points on a Coordinate Grid

In coordinate geometry, one of the fundamental concepts is finding the midpoint between two given points on the Cartesian plane. This process is essential in various real-life scenarios, such as navigation, architecture, and digital mapping.

Let’s explore a practical situation where a boy named Billy is trying to find the exact midpoint between his house and his best friend’s house. Understanding this midpoint can help them meet halfway, plan routes, or just explore locations in between!

📍 Understanding the Coordinate Grid

The coordinate grid, also known as the Cartesian plane, consists of a horizontal axis (x-axis) and a vertical axis (y-axis) intersecting at the origin point (0, 0). Each location or point on this plane is defined using a pair of values (x, y).

    In this scenario:
    Billy’s house is located at: (8, 7)
His best friend’s house is at: (2, -1)

  

📐 What is a Midpoint?

The midpoint is the point that is exactly halfway between two coordinates on the plane. It lies directly in the middle of a line segment connecting the two points.

To find it, we use a standard geometric formula called the Midpoint Formula.

Midpoint = \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)

Where:

\( x_1, y_1 \) are the coordinates of the first point
\( x_2, y_2 \) are the coordinates of the second point

🧮 Step-by-Step Midpoint Calculation

Step 1: Identify the Coordinates

First, we write down the two coordinates:

Point A (Billy’s House): (8, 7)
Point B (Friend’s House): (2, -1)

Step 2: Apply the Midpoint Formula

Plug the values into the formula:

Midpoint = \( \left( \frac{8 + 2}{2}, \frac{7 + (-1)}{2} \right) \)

Step 3: Simplify Each Component

Midpoint = \( \left( \frac{10}{2}, \frac{6}{2} \right) \)

Simplify further:

Midpoint = (5, 3)

✅ Final Answer: The midpoint between the two houses is located at (5, 3).

📊 Visual Interpretation

Imagine plotting both houses on a coordinate plane. Billy’s house is in the first quadrant (positive x and y values), and his friend’s house is in the fourth quadrant (positive x, negative y). The line connecting these two points slopes downward.

The midpoint at (5, 3) lies directly between them. It is closer to the origin than Billy’s house and slightly higher than his friend’s house.

📈 Practical Use of the Midpoint

Knowing the midpoint is useful in various ways:

📍 It can serve as a meeting location that is fair and equidistant.
🛣️ Helps in planning routes, especially when minimizing travel distance.
🏫 Could represent the location for a new service (like a cafe or school) that should be centrally located.
🧮 Used frequently in programming for drawing and animation.

🧠 Deep Dive into the Midpoint Concept

The midpoint formula is derived from the concept of averaging values. You are simply taking the average of the x-coordinates and the y-coordinates:

X-component: \( \frac{8 + 2}{2} = 5 \)
Y-component: \( \frac{7 + (-1)}{2} = 3 \)

The result, (5, 3), is guaranteed to lie on the segment joining the two original points. In fact, if you were to calculate the distance from (5, 3) to both houses using the distance formula, the values would be equal.

🧮 Quick Bonus: Distance from Midpoint to Either House

Let’s verify that the midpoint is equidistant by using the distance formula:

Distance = \( \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2} \)

Example: From midpoint (5, 3) to Billy’s house (8, 7):

\( \sqrt{(8 – 5)^2 + (7 – 3)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \)

From midpoint (5, 3) to friend’s house (2, -1):

\( \sqrt{(5 – 2)^2 + (3 + 1)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \)

🔁 The distance from midpoint to each house is equal. This confirms that (5, 3) is truly the center point between the two homes.

📚 Final Thoughts

Mastering the midpoint formula opens doors to a deeper understanding of coordinate geometry. Whether you’re a student, a math enthusiast, or someone using maps and spatial data, this simple yet powerful formula can assist with countless real-life applications.

In our example with Billy and his friend, the coordinate (5, 3) perfectly represents the halfway point on the grid, offering a mathematically fair solution for where the two could meet — halfway between their homes.

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