Problem 38 - Algebra 1

Question

Find the midpoint between the two points \((-3,3),(2,-2)\)

Step-by-Step Solution

Answer

The midpoint is \(\left(-0.5, 0.5\right)\)

Step 1: Identify the coordinates of the points

The given points are \((-3,3)\) and \((2,-2)\), so we have \(x1 = -3, y1 = 3, x2 = 2, y2 = -2\)

Step 2: Insert the coordinates into the formula

Substitute the values of \(x1, y1, x2, y2\) into the midpoint formula. We get \(M = \left( \frac{-3+2}{2} , \frac{3+(-2)}{2} \right)\)

Step 3: Calculate the coordinates of the midpoint

Perform the calculations which gives us the midpoint \(M = \left( \frac{-1}{2} , \frac{1}{2} \right)\)

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that establishes a connection between algebra and geometry. This field utilizes a coordinate system to relate points on a plane through numbers and equations. It allows us to answer geometric questions with algebraic equations. Thanks to coordinate geometry, we can pinpoint exact positions on a plane using coordinates and analyze shapes through equations.

• Visualizing shapes and lines with coordinates.
• Translating geometric problems into algebraic models.

Coordinate geometry is essential for understanding the physical world and solving complex problems using straightforward mathematical techniques.

Cartesian Coordinates

The Cartesian coordinate system is a two-dimensional system used to determine the position of points. It is named after the French mathematician René Descartes, and it revolutionized geometry by introducing a link between algebra and geometry, where each point is defined by a pair of numerical coordinates:

• The point \((x, y)\) consists of an "x" coordinate (horizontal) and a "y" coordinate (vertical).
• The origin, where \(x = 0\) and \(y = 0\), is the center point from which all calculations are made.

To illustrate, a point like \((-3, 3)\) denotes 3 units up along the y-axis and 3 units left along the x-axis. The simplicity and utility of Cartesian coordinates make them indispensable for solving geometric problems, like finding the midpoint.

Distance and Midpoints

The concepts of distance and midpoints are fundamental in coordinate geometry. The midpoint formula helps us find the exact middle point between two coordinates by averaging the x and y values of the initial points. Using the coordinates from the example, given \((-3, 3)\) and \(2, -2)\):

• Midpoint Formula: \((\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})\)
• Calculate: \((-3 + 2) / 2 = -1/2\) and \((3 - 2) / 2 = 1/2\)

The midpoint of these points is \((-1/2, 1/2)\), dividing the distance between them equally. Depending on context, understanding the midpoint can be critical, such as when trying to find balance points in physical systems or determining halfway markers in data analysis.