Problem 24 - Algebra 2

Question

Find the midpoint of the line segment with endpoints at the given coordinates. Then find the distance between the points. $$ \left(-3,-\frac{2}{11}\right),\left(5, \frac{9}{11}\right) $$

Step-by-Step Solution

Answer

Midpoint: (1, 7/22); Distance: \(\sqrt{65}\).

Step 1: Identify the coordinates

First, identify the coordinates of the two points involved. They are given as \((-3, -\frac{2}{11})\) and \((5, \frac{9}{11})\).

Step 2: Calculate the midpoint

The formula to find the midpoint \(M\) of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \].Substitute the values \(x_1 = -3\), \(y_1 = -\frac{2}{11}\), \(x_2 = 5\), and \(y_2 = \frac{9}{11}\) into the formula:\[ M = \left( \frac{-3 + 5}{2}, \frac{-\frac{2}{11} + \frac{9}{11}}{2} \right) \].This simplifies to:\[ M = \left( \frac{2}{2}, \frac{7}{11 \times 2} \right) \] = \( (1, \frac{7}{22}) \).

Step 3: Use the distance formula

Use the distance formula to find the distance \(d\) between the points. The formula is:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \].Substitute the values:\[ d = \sqrt{(5 - (-3))^2 + (\frac{9}{11} - (-\frac{2}{11}))^2} \].Simplifying, this becomes:\[ d = \sqrt{(5 + 3)^2 + (\frac{9}{11} + \frac{2}{11})^2} \] = \[ \sqrt{8^2 + \frac{11^2}{11^2}} \] = \[ \sqrt{64 + \left(1\right)} \] = \[ \sqrt{65} \].

Step 4: Simplify the results

The midpoint is \( (1, \frac{7}{22}) \), and the distance is \( \sqrt{65} \). The square root \( \sqrt{65} \) is left in the simplest radical form, as it cannot be simplified further.

Key Concepts

Coordinate Geometry

Coordinate Geometry is the study of geometry using a coordinate system. Think of the XY grid with horizontal and vertical lines as the foundation. Points are plotted using numerical values called coordinates. These coordinates are written as ordered pairs \(x, y\), where \(x\) is the position along the horizontal axis, and \(y\) is the vertical axis's position.

The beauty of coordinate geometry lies in its power to describe geometric entities and solve geometry problems using algebra. By simply using numbers, we can accurately locate points, analyze lines, and calculate distances and midpoints.

In this specific exercise, points had coordinates \((-3, -\frac{2}{11})\) and \(5, \frac{9}{11})\) on a grid, making them easy to work with using formulas like the Distance Formula and Midpoint Formula.

Distance Formula

The Distance Formula is a valuable tool in coordinate geometry for determining the length between two points on a line. If you imagine the points connected by a straight line segment, the formula tells you how long this segment is. It is derived from the Pythagorean Theorem, reflecting how geometry and algebra beautifully intersect.

The formula is:

• \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Example: Suppose you have the points \( (x_1, y_1) = (-3, -\frac{2}{11}) \) and \( (x_2, y_2) = (5, \frac{9}{11}) \).

Let's substitute them into the formula:

• Find each difference: \(x_2 - x_1 = 5 - (-3) \)
• \(y_2 - y_1 = \frac{9}{11} - (-\frac{2}{11}) \)

Then, with these differences calculated, the next step is:

• Square them individually.
• Add those squares together.
• Finally, take the square root of that sum.

This process results in: \( \sqrt{65} \), the precise distance between the two points.

Midpoint Formula

The Midpoint Formula helps us find the center point of a line segment connecting two given points in coordinate geometry. It's a handy tool to ensure symmetrical work when dividing a segment into equal parts. Think of the midpoint as the average position between two endpoints.

The formula for the midpoint M is:

• \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

This formula calculates the average of the x-coordinates and the y-coordinates of the endpoints, giving us the midpoint.

Given the points \( (-3, -\frac{2}{11}) \) and \( (5, \frac{9}{11}) \):

• Find the average of the x-coordinates: \( \frac{-3 + 5}{2} \)
• Find the average of the y-coordinates: \( \frac{-\frac{2}{11} + \frac{9}{11}}{2} \)

By computing these averages, we derive the coordinates of the midpoint: \( (1, \frac{7}{22}) \).

Line Segment

A line segment in geometry is a part of a line that is enclosed between two endpoints. Unlike a line that extends infinitely in both directions, a line segment has a fixed start and end.

Understanding line segments is crucial in coordinate geometry. They form the building blocks for more complex geometric shapes and allow us to apply tools like the Distance and Midpoint Formulas.

In this scenario, the line segment connects points \( (-3, -\frac{2}{11}) \) and \( (5, \frac{9}{11}) \). By locating these endpoints on the grid, you can visually inspect the segment and employ the coordinate geometry formulas discussed.

The midpoint divides this segment into two equal halves, while the distance formula measures its overall length. Each concept helps in the comprehensive analysis of the segment's properties, providing valuable insights into the vector's characteristics.