Problem 28 - 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. \((2,7),(4,3)\)

Step-by-Step Solution

Answer

The Midpoint of the two points in the given exercise is (3,5) and the distance from each point to the midpoint is \(\sqrt{5}\)

Step 1: Calculate the Midpoint

The coordinates of our two points are (2,7) and (4,3). We substitute these into the midpoint formula:\n Midpoint = \((\frac{2+4}{2}, \frac{7+3}{2}) = (3,5)\)

Step 2: Calculate the Distance between Point 1 and the Midpoint

We proceed to calculate the distance between the first point (2,7) and the Midpoint(3,5) using the distance formula:\n Distance = \(\sqrt{(3-2)^2+(5-7)^2} = \sqrt{1+4} = \sqrt{5}\)

Key Concepts

Distance Formula

The distance formula is a fundamental idea in coordinate geometry. It helps us determine the distance between two points in a coordinate plane. Imagine you have two points, \( (x_1,y_1) \) and \( (x_2,y_2) \), forming a diagonal line when plotted. The distance formula comes from the Pythagorean theorem, which effectively measures straight-line distance.
The formula is:\[Distance = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Here, \( (x_2-x_1) \) represents the horizontal gap, and \( (y_2-y_1) \) the vertical gap. Squaring these differences helps us deal with negative values and get a positive distance. This step-by-step process provides a practical way to find precise distances between any two given points.

Coordinate Geometry

Coordinate geometry, also called analytic geometry, involves using a coordinate plane to understand and solve geometric problems. While traditional geometry is primarily about shapes and spaces, coordinate geometry gives these visual ideas a numerical basis. This makes it easier to apply algebraic methods.In a coordinate plane, each point has unique coordinates \( (x, y) \). The "x" marks its position relative to the horizontal axis, and the "y" denotes its spot against the vertical axis.

Some of the main tools in coordinate geometry include:

• The distance formula to find the distance between points.
• The midpoint formula to determine the center point of a line segment.
• Equations of lines, which illustrate relationships between points.

By leveraging these tools, you can resolve problems involving line segments, shapes, and journeys, turning visual tasks into numerical calculations.

Line Segments

Line segments are fundamental components of geometry that define the part of a line bounded by two endpoints. In the context of coordinate geometry, they often link two points and can be analyzed using various formulas, like the distance formula for finding their length.
A line segment has:

• Endpoints: Where it starts and ends, each defined by coordinates.
• Length: Found using the distance between endpoints.
• Midpoint: Found using the midpoint formula to discover its center.

In the provided example, finding the midpoint ensures that both endpoints are equidistant from this center point. Utilizing the midpoint formula helps confirm that the middle is exactly halfway between, ensuring balance in the line segment's length distribution. Understanding line segments deeply aids in grasping larger geometric concepts, as they are often structural elements or building blocks in more complex figures.