Problem 25 - 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. \((1,2),(0,0)\)

Step-by-Step Solution

Answer

The midpoint of the line segment connecting the points \((1,2)\) and \((0,0)\) is \((0.5, 1)\), and the distance from this midpoint to each of the original points is \(\sqrt{1.25}\), hence proving that the midpoint is equally distant from each point.

Step 1: Calculate the midpoint

Using the midpoint formula, the midpoint (M) of the two points \((1,2)\) and \((0,0)\) can be calculated as: M = \(((1+0)/2, (2+0)/2) = (0.5, 1)\)

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

Using the distance formula, the distance from the mid-point to the first point \((1,2)\) can be calculated as: d1 = \(\sqrt{(1-0.5)^2 + (2-1)^2} = \sqrt{(0.5)^2 + (1)^2} = \sqrt{0.25+1} = \sqrt{1.25}\)

Step 3: Calculate the distance from the midpoint to the second point

Using the distance formula, the distance from the midpoint to the second point \((0,0)\) can be calculated as: d2 = \(\sqrt{(0.5-0)^2 + (1-0)^2} = \sqrt{(0.5)^2 + (1)^2} = \sqrt{0.25+1} = \sqrt{1.25}\)

Step 4: Compare the distances

It can be seen that both distances calculated in Steps 2 and 3 are equal. Therefore, the midpoint is the same distance from each point.

Key Concepts

Distance Formula

The distance formula is a crucial tool in coordinate geometry, often used to find the distance between two points on a Cartesian plane. It's derived from the Pythagorean theorem. Given two points \((x_1, y_1)\) and \((x_2, y_2)\), the distance \(d\) between these points is calculated as:

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

In the context of the exercise, we use this formula to verify that the midpoint is equidistant from both given points, demonstrating symmetry and balance across the midpoint of a line segment.
The distance formula helps ensure our answer makes sense, as it confirms the midpoint is the true center of the line segment.

Line Segment

A line segment is a part of a line that is bounded by two distinct endpoints. It has a fixed length, unlike a line that extends infinitely in both directions.
In the coordinate plane, a line segment can be represented by its endpoints, such as \((x_1, y_1)\) and \((x_2, y_2)\). In this example, our endpoints are \((1,2)\) and \((0,0)\).
To understand the midpoint and distance better, visualizing the line segment helps to see the symmetry and the division of the segment into two equal parts by the midpoint. The midpoint divides a line segment into two equal lengths. So, not only is the
midpoint essential in ensuring symmetry, but it also helps us understand the properties of the line segment being bisected.

Coordinate Geometry

Coordinate geometry, or analytic geometry, involves the study of geometric figures using the coordinate plane. By placing geometric shapes in this numerical system, we can use algebraic principles to explore their properties.

• It uses coordinates to solve geometric problems.
• This approach makes calculating shapes, lengths, and distances systematic.

For example, in the exercise, coordinate geometry allows us to apply formulas like the midpoint and distance formulas to find specific points and confirm equal lengths.
This method simplifies the process of proving the midpoint is equidistant from the points of a line segment, bringing together algebra and geometry seamlessly. Ultimately, coordinate geometry provides a powerful framework channel that merges numerical and spatial reasoning.