Problem 4 - Algebra and Trigonometry

Question

The point midway between \((a, b)\) and \((c, d)\) is ________. So the point midway between \((1,2)\) and \((7,10)\) is ________.

Step-by-Step Solution

Answer

The point midway between \((1,2)\) and \((7,10)\) is \((4,6)\).

Step 1: Understand the Concept

The midpoint of two points in a coordinate plane is calculated by averaging the x-coordinates and the y-coordinates of the points respectively.

Step 2: Midpoint Formula

The formula for finding the midpoint M between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Step 3: Plug in the First Pair of Coordinates

Substitute \( a = 1, b = 2, c = 7, d = 10 \) into the midpoint formula. \[ M = \left( \frac{1 + 7}{2}, \frac{2 + 10}{2} \right) \]

Step 4: Simplify the Expressions

Calculate the averages for the x-coordinates and y-coordinates: \( x = \frac{1 + 7}{2} = 4 \) \( y = \frac{2 + 10}{2} = 6 \)

Step 5: Write the Midpoint Coordinate

Combine the calculated x and y values to get the midpoint coordinate: The point midway between \((1,2)\) and \((7,10)\) is \((4, 6)\).

Key Concepts

Coordinate Plane

A coordinate plane is a two-dimensional surface where each point is defined by a pair of numerical coordinates. These coordinates are often called the x-coordinate and y-coordinate and they specify the point's position relative to the horizontal and vertical axes.
Think of it as a giant grid with horizontal and vertical lines, intersecting at zero. That's where the magic happens! It allows us to precisely talk about where things are. When working with the coordinate plane, you'll often map out two points and determine the distance or calculate something like a midpoint between these points.
It's crucial for solving problems in geometry or any other subject needing precise spatial understanding.

Averaging Coordinates

To find something in between two numbers on the coordinate plane, like a midpoint, you average the numbers.
Averaging is simple: you add the numbers together, then divide by 2. This gives you the number exactly halfway between them. For coordinates, you must do this both for the x-coordinates and for the y-coordinates separately.
Imagine you have points \( (x_1, y_1) \) and \( (x_2, y_2) \). To find the average:

• For x: \( \frac{x_1 + x_2}{2} \)
• For y: \( \frac{y_1 + y_2}{2} \)

By practicing with a few examples, you'll start to understand how these averages really help in finding midpoints and understanding the geometry of the plane.

Midpoint Calculation

The midpoint is like a balancing point between two endpoints on the coordinate plane. It represents the exact center of a segment connecting two distinct points.
To find the midpoint, we use the midpoint formula: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]This formula might look intimidating, but it's just putting together the simple averages of the x and y coordinates. Drawing these out visually can help too.
For our example:

• You had the points \( (1, 2) \) and \( (7, 10) \).
• Find the average of the x-coordinates: \( \frac{1 + 7}{2} = 4 \).
• Find the average of the y-coordinates: \( \frac{2 + 10}{2} = 6 \).

Putting it together, the midpoint is \( (4, 6) \). This formula helps you quickly find the center spot between two places on a grid.