Problem 31 - Algebra 1: Concepts and Skills

Question

Find the midpoint of the line segment connecting the given points. Then show that the midpoint is the same distance from each point. \((-9,17),(5,-7)\)

Step-by-Step Solution

Answer

The midpoint of the line segment is (-2, 5) and it is equidistant from the two points.

Step 1: Calculate the midpoint

To find the midpoint of the line segment, average the x-coordinates and the y-coordinates of the two points. The formula for the midpoint is \((\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})\). So the midpoint M of the line segment connecting the points (-9,17) and (5,-7) is \((\frac{-9+5}{2},\frac{17+(-7)}{2})\) which simplifies to \((-2,5)\).

Step 2: Calculate the distance from each point to the midpoint

To verify that the midpoint is equidistant from the given points, we calculate the Euclidean distance from each point to the midpoint using the distance formula, \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\). The distance D1 from the point (-9,17) to the midpoint (-2,5) is \(\sqrt{(-2-(-9))^2+(5-17)^2}\). Simplify this to get \(\sqrt{7^2+(-12)^2}\), which is \(\sqrt{193}\). The distance D2 from (5,-7) to (-2,5) is \(\sqrt{(-2-5)^2+(5-(-7))^2}\), which simplifies to \(\sqrt{7^2+12^2}\), that is also equal to \(\sqrt{193}\).

Step 3: Show the distances are equal

Since both distances D1 and D2 are equal i.e. \(\sqrt{193}\), it confirms that the midpoint is the same distance from each of the given points.

Key Concepts

Distance Formula

The Distance Formula is a key tool in geometry, especially when you're dealing with points on a coordinate plane. It helps you find the length of the line segment connecting two points by measuring the "as-the-crow-flies" distance between them.

To calculate the distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \), you use the formula:

• \( \sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \)

This formula is derived from the Pythagorean Theorem, which applies to right triangles. Imagine a right triangle where the line segment is the hypotenuse, and the differences in x and y coordinates form the legs.

When you plug in numbers, like for the point calculations in our original exercise, it's essentially finding out the change in x and change in y, squaring them, adding them up, and then taking the square root. This gives you a direct path to how far one point is from another.

Coordinate Geometry

Coordinate Geometry, also known as analytic geometry, merges algebra with geometry and gives you the ability to use equations and formulas on a spatial plane. This field introduces a coordinate system where every point can be defined numerically by its x and y values.

You're able to

• Calculate distances (using the Distance Formula)
• Find midpoints
• Determine slopes of lines

These calculations aren't just theoretical. They're useful in many practical situations such as navigation, architecture, and computer graphics.

The power of coordinate geometry lies in its ability to take abstract geometric figures and deal with them algebraically. You can prove geometric theorems and solve real-world problems quite effectively. In your practice, you worked with two points to find the midpoint by averaging their x and y values, a direct application of this geometric approach.

Linear Equations

Linear Equations are fundamental in both mathematics and everyday applications. They describe a straight line on the coordinate plane and are often expressed in the form \( y = mx + c \), where \( m \) is the slope, and \( c \) is the y-intercept.

Linear equations can help you make predictions and interpretations based on data. Given two points, you can determine the equation of the line that passes through these points by:

• Calculating the slope (which is the rise over run, or \( \frac{y_2-y_1}{x_2-x_1} \))
• Substituting a point into the equation to solve for the intercept

In problems involving midpoints and distances, linear equations can be used to understand the broader context of data and relationships.

While our current exercise isn't focused on finding a specific linear equation, recognizing how the points, distances, and midpoints relate to lines can lead to a deeper understanding of linear relationships in a variety of settings.