Problem 20 - College Algebra

Question

Find the midpoint of each line segment with the given endpoints. $$(10,4)\( and \)(2,6)$$

Step-by-Step Solution

Answer

The midpoint of the line segment with endpoints (10, 4) and (2, 6) is (6, 5).

Step 1: Identify the given points

The given points are A(10,4) and B(2,6). The coordinates of point A are \(x_1 = 10, y_1 = 4\) and the coordinates of point B are \(x_2 = 2, y_2 = 6\)

Step 2: Apply the midpoint formula

The midpoint formula is given by: \[M = \(\frac{{x_1 + x_2}}{2}, \frac{{y_1 + y_2}}{2}\)\]Where M is the midpoint. So, substitute the values of \(x_1, x_2, y_1, y_2\) into the formula and compute the two values.

Step 3: Calculate the midpoint

The coordinates of the midpoint M can be found by substituting the coordinates of A and B into the midpoint formula:\[M = \(\frac{{10 + 2}}{2}, \frac{{4 + 6}}{2}\) = (6, 5)\]So, the coordinates of the midpoint are (6, 5).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This method enables calculations and predictions through the use of algebraic formulas to address geometric problems.
In coordinate geometry, points are located on the 'coordinate plane' using an ordered pair of numbers known as coordinates. The coordinate plane has two axes: the horizontal axis is called the x-axis, and the vertical axis is called the y-axis. Each point in the plane is defined by an (x, y) pair, where 'x' represents the position relative to the x-axis, and 'y' represents the position relative to the y-axis.

• Axes: Two perpendicular lines on the plane.
• Coordinates: An ordered pair of numbers (x, y).
• Quadrants: The x and y axes divide the plane into four quadrants.
• Origin: The point (0, 0) where the x and y axes intersect.

Utilizing this approach, we can measure distances and establish midpoints, slopes, and intersections, transforming geometric problems into algebraic equations that are often simpler to solve.

Algebraic Formulas

Algebraic formulas are equations that relate algebraic expressions to solve for unknown variables. They serve as the backbone of algebra and are used to tackle a wide range of mathematical problems. Among these, the midpoint formula is particularly important in coordinate geometry.

Some of the common algebraic formulas include the quadratic formula used to solve quadratic equations, the formulas for the sum and product of roots, and, pertinent to this discussion, the midpoint formula. This formula is a means of finding a point that is exactly halfway between two given points on a coordinate plane. Recognizing and applying algebraic formulas correctly is crucial for solving problems efficiently and accurately. They not only provide a methodical approach to finding solutions but also help in understanding the structure and relationships between different mathematical concepts.

Examples of Algebraic Formulas:

• The equation of a line: \(y = mx + b\)
• Slope formula: \(m = \frac{y2 - y1}{x2 - x1}\)
• Midpoint formula: \(M = \left(\frac{{x1 + x2}}{2}, \frac{{y1 + y2}}{2}\right)\)

Finding Midpoints

Finding the midpoint of a line segment in coordinate geometry involves determining the point that is equidistant from the endpoints of the segment. The midpoint is found using the midpoint formula: \(M = (\frac{{x1 + x2}}{2}, \frac{{y1 + y2}}{2})\).
This formula takes the average of the x-coordinates and the average of the y-coordinates of the endpoints to find the exact halfway point between them. When computing this by hand, simply add the x-coordinates of the two endpoints together and divide by two, and do the same with the y-coordinates.
Let's see this in an applied exercise:
You have the coordinates of two points, A(10, 4) and B(2, 6). Applying the midpoint formula, we add the x-coordinates (10 + 2) and the y-coordinates (4 + 6) of A and B, and then divide each sum by 2, resulting in the coordinates (6, 5) as the midpoint M. Thus, M(6, 5) is the exact center point between A and B on the coordinate plane. This process demonstrates an elegant intersection between algebra and geometry where formulas facilitate the calculation of key geometric figures.