Question

What is the midpoint between \((-2,-3)\) and \(\left(1, \frac{1}{2}\right) ?\) \(\begin{array}{llll}\text { (A) }\left(-1,-2 \frac{1}{2}\right) & \text { (B }\left(-\frac{1}{2},-2 \frac{1}{2}\right) & \text { C }\left(-1,-1 \frac{1}{4}\right) & \text { \odot }\left(-\frac{1}{2},-1 \frac{1}{4}\right)\end{array}\)

Short answer

The midpoint between \((-2,-3)\) and \(\left(1, \frac{1}{2}\right)\) is \(\left(-\frac{1}{2}, -1\frac{1}{4}\right)\).

Step-by-step solution

Identify the Coordinates of the Two Points

The two points provided are \((-2,-3)\) and \(\left(1, \frac{1}{2}\right)\) . So, \(x_1 = -2\), \(y_1 = -3\), \(x_2 = 1\), and \(y_2 = \frac{1}{2}\).

Calculate the Midpoint's X-Coordinate

Using the midpoint formula \(x = \frac{x_1 + x_2}{2}\), the x-coordinate of the midpoint is \(x = \frac{-2 + 1}{2} = -\frac{1}{2}\).

Calculate the Midpoint's Y-Coordinate

Similarly, for the y-coordinate, apply the midpoint formula \(y = \frac{y_1 + y_2}{2}\), that gives \(y = \frac{-3 + \frac{1}{2}}{2} = -\frac{5}{4} = -1\frac{1}{4}\).

Conclude the Final Answer

Collecting the x and y coordinates, the midpoint of \((-2,-3)\) and \(\left(1, \frac{1}{2}\right)\) is \(-\frac{1}{2}, -1\frac{1}{4}\).

Key concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic equations to describe and analyze points, lines, and shapes on a coordinate plane. The coordinate plane is a two-dimensional surface defined by a horizontal axis, known as the x-axis, and a vertical axis, called the y-axis. Points on this plane are typically represented by pairs of numbers \( (x, y) \), where \( x \) and \( y \) correspond to the position on the x-axis and y-axis, respectively.

In coordinate geometry, the relationship between two points can be described using formulas. These points theorems, lines equation and distance or midpoint formulas are fundamental tools used to find the attributes of geometric figures within the plane. For instance, when you're working with midpoints, you'd use the midpoint formula to determine the exact central point between two other points on the plane, providing insights into the geometry of a line segment.

Algebraic Concepts

The field of algebra involves studying symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. One of the core algebraic concepts is the use of equations to represent relationships between variables. When dealing with coordinate geometry, algebra allows us to succinctly express complex geometric concepts, like the midpoint of a line segment, through simple formulas.

Understanding the Formula's Components

For example, in the midpoint formula \( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \), \(x_1\) and \(y_1\) are the coordinates of the first point, while \(x_2\) and \(y_2\) are those of the second. The numerators \(x_1 + x_2\) and \(y_1 + y_2\) add the coordinates of the points, effectively combining their horizontal and vertical positions. Dividing by two simply averages these positions, placing the midpoint directly between the original points on both axes.

Algebraic manipulation is essential in transforming formulas and making them applicable to specific problems, which allows students to find solutions through methodical processes rather than rote memorization.

Solving Midpoint Problems

Solving midpoint problems in coordinate geometry requires both an understanding of the midpoint formula as well as careful algebraic manipulation. The midpoint formula, \( \left( \frac{x_1 + x_2}{2} , \frac{y_1 + y_2}{2} \right) \), is a practical application of algebraic concepts in coordinate geometry, which takes the coordinates of two points and calculates the coordinates of the point that lies halfway between them.

Applying the Midpoint Formula

When you're presented with a question asking for the midpoint, like the exercise here, follow a systematic approach:

• Identify the x and y coordinates of each given point.
• Insert these values into the midpoint formula.
• Simplify the expressions to find the x and y coordinates of the midpoint.
• Combine the x and y coordinates into an ordered pair that represents the midpoint's location on the coordinate plane.

Through this methodology, the application of the midpoint formula transitions from a rote memorization of steps into a clear, logical process that reinforces the interconnectedness of algebra and geometry on the coordinate plane.