Problem 25 - Algebra 2

Question

Find the midpoint of the line segment with endpoints at the given coordinates. Then find the distance between the points. $$ \left(0, \frac{1}{5}\right),\left(\frac{3}{5},-\frac{3}{5}\right) $$

Step-by-Step Solution

Answer

Midpoint: \( \left( \frac{3}{10}, -\frac{1}{5} \right) \), Distance: 1.

Step 1: Identifying Coordinates

First, identify the coordinates of the endpoints. We have the points \( A(0, \frac{1}{5}) \) and \( B(\frac{3}{5}, -\frac{3}{5}) \).

Step 2: Calculating the Midpoint Formula

The midpoint \( M(x, y) \) of a segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \].

Step 3: Substitute and Discuss Midpoint

Substitute the given coordinates into the midpoint formula:\[ x = \frac{0 + \frac{3}{5}}{2} = \frac{3}{10} \], and \[ y = \frac{\frac{1}{5} + (-\frac{3}{5})}{2} = \frac{-\frac{2}{5}}{2} = -\frac{1}{5} \].So the midpoint is \( M = \left( \frac{3}{10}, -\frac{1}{5} \right) \).

Step 4: Distance Formula

The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \].

Step 5: Substitute and Calculate Distance

Substitute the coordinates into the distance formula:\[ d = \sqrt{\left(\frac{3}{5} - 0\right)^2 + \left(-\frac{3}{5} - \frac{1}{5}\right)^2} = \sqrt{\left(\frac{3}{5}\right)^2 + \left(-\frac{4}{5}\right)^2} \].Calculate:\[ d = \sqrt{\frac{9}{25} + \frac{16}{25}} = \sqrt{\frac{25}{25}} = \sqrt{1} = 1 \].

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This branch of mathematics allows us to find and analyze the geometric properties of figures on a plane using algebraic equations. By placing geometric figures in a coordinate plane, we can use numerical methods to solve problems like finding the midpoint of a segment or the distance between two points.

If we take the line segment defined by points in the plane, its endpoints have coordinates that can be identified on the coordinate plane. This approach simplifies many geometric problems and makes it possible to use algebraic techniques to gain insights into geometry.

• It helps us visualize mathematical concepts in a graphical format.
• Utilizes x and y coordinates to determine the position of points on a plane.
• Enables the derivation of precise geometric properties using algebraic formulas.

Coordinate geometry is fundamental in mathematics because it merges algebra with geometry, providing a powerful tool to solve problems in a simple and structured way.

Distance Between Points

The distance between two points in a coordinate plane is essentially the length of the straight line segment that connects them. This is known as the distance formula, which is derived from the Pythagorean Theorem.

To find the distance between two points \(x_1, y_1\) and \(x_2, y_2\), we use the formula:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \].

This formula calculates the hypotenuse of a right triangle, where the horizontal and vertical differences represent the legs of the triangle.

• Calculate the difference between the x-coordinates and the y-coordinates of the two points.
• Square each of these differences.
• Add these squared differences together, then take the square root of the resulting sum.

This method gives you the absolute distance between any two points on the plane. It's a fundamental concept in both mathematics and real-world applications, such as navigation and measuring spatial distances.

Midpoint of a Segment

The midpoint of a line segment is a point that divides the segment into two equal parts. It is found by taking the average of the x-coordinates and the y-coordinates of the endpoints of the segment.

If the endpoints are denoted by \(x_1, y_1\) and \(x_2, y_2\), the midpoint \( M(x, y) \) can be calculated as:\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \].

This formula works because:

• It averages the x-values, giving the x-coordinate of the midpoint.
• It averages the y-values, giving the y-coordinate of the midpoint.

By computing the midpoint, you effectively find the center of the segment, making it useful in various geometric constructions and ensuring symmetry. Understanding how to find a midpoint helps when you need to bisect a segment or find center points for geometric shapes.