Problem 10 - Algebra 1

Question

Find the midpoint between the two points. \((4,4),(-1,2)\)

Step-by-Step Solution

Answer

The midpoint between the points \((4,4)\) and \((-1,2)\) is \((1.5, 3)\)

Step 1: Recognize the given points

The given points are \((4,4)\) and \((-1,2)\). Here, \(x_1 = 4\), \(y_1 = 4\), \(x_2 = -1\), and \(y_2 = 2\).

Step 2: Apply the formula

Apply the midpoint formula \(\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\) to the given points. This gives us \(\left(\frac{4+(-1)}{2}, \frac{4+2}{2}\right)\).

Step 3: Simplify the result

Simplify the result which gives the point \(\left(\frac{3}{2}, 3\right)\) or \((1.5, 3)\).

Key Concepts

Understanding Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the process of plotting and describing geometrical figures using a coordinate system. In this framework, geometric shapes are described algebraically using coordinates on a plane. This helps us understand how different points relate to each other in terms of position.

• The x-axis (horizontal line) and y-axis (vertical line) are the two components of a coordinate plane.
• Each point on this plane is defined by an ordered pair \(x, y\) that denotes its exact position relative to the origin (0, 0).

Coordinate geometry allows us to calculate various geometric properties like distance, slope, and, as in this case, the midpoint between two points. By transforming geometric problems into numerical ones, it provides a systematic approach to analysis.

How to Find the Midpoint

Finding the midpoint between two points is all about averaging the x-coordinates and the y-coordinates of those points separately. The midpoint formula \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\) helps in determining the center point between two given coordinates, effectively dividing the line connecting these points into two equal sections.

• Step-by-Step:
• For points (4, 4) and (-1, 2), substitute their coordinates into the midpoint formula.
• Calculate \(\frac{4 + (-1)}{2}\) to find the average of the x-coordinates.
• Calculate \(\frac{4 + 2}{2}\) to find the average of the y-coordinates.
• The resulting midpoint is \(\left(\frac{3}{2}, 3\right)\), which is equivalent to (1.5, 3).

This method provides a practical way to determine precisely where a midpoint is located between any two locations on a coordinate plane.

Applying Algebra in Geometry Problems

Algebra plays a significant role in solving geometric problems, providing a bridge between geometric figures and their numerical representations. When you're asked to find the midpoint or any other property in coordinate geometry, algebra simplifies the process.

• Using Algebraic Formulas: Instead of relying solely on visual representations, algebraic formulas help in performing precise calculations required to find geometric properties.
• The midpoint formula is an algebraic expression essential in determining symmetry and balance between two endpoints.
• Substitution and simplification are key algebraic techniques used to solve coordinate geometry problems, as shown in finding the midpoint between two points.

Thus, algebra is not just about symbols and formulas; it's a powerful tool in clarifying and solving real-world geometric problems effectively.