Problem 39 - Algebra 1

Question

Find the midpoint between the two points \((2,7),(4,3)\)

Step-by-Step Solution

Answer

The midpoint between the points \((2,7)\) and \((4,3)\) is \((3,5)\).

Step 1: Identify the coordinates

First, identify the coordinates of the two given points. They are \((x_1, y_1) = (2,7)\) and \((x_2, y_2) = (4,3)\).

Step 2: Apply the midpoint formula

The formula for finding the midpoint between two points is \((\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})\). Substitute the values of \(x_1\), \(x_2\), \(y_1\), and \(y_2\) into the formula. This will give: \((\frac{2+4}{2}, \frac{7+3}{2})\).

Step 3: Calculate the midpoint

Now, calculate the midpoint. This will give: \((\frac{6}{2}, \frac{10}{2})\) or equivalently \((3,5)\).

Key Concepts

Coordinate Geometry

Coordinate geometry, often referred to as analytical geometry, is a crucial branch of mathematics that deals with the study of geometry using a coordinate system. This method allows us to solve geometric problems in a more algebraic way by using coordinates. In this context, points are placed in specific locations on the plane, which can then be expressed using algebraic formulas. This dual approach offers a powerful way to bridge the visual aspect of geometry with the precision of algebra.
In coordinate geometry, each point is defined by an ordered pair

• The first number, commonly called the x-coordinate, indicates the distance along the horizontal axis.
• The second number, or y-coordinate, shows the distance along the vertical axis.

The main advantage of coordinate geometry is its ability to convert geometric shapes into equations. For example, lines and curves can be expressed using equations, which makes it easier to analyze and solve problems involving them. By utilising these equations, we can derive important information such as slopes, distances, and midpoints.

Cartesian Plane

The Cartesian plane is a two-dimensional plane created by two number lines intersecting at a right angle. These number lines are known as axes. The horizontal line is called the x-axis, and the vertical one is the y-axis. Their intersection point, known as the origin, is where both axis values are zero—denoted as (0, 0).
The Cartesian plane is divided into four quadrants:

• Quadrant I: Both x and y are positive.
• Quadrant II: x is negative and y is positive.
• Quadrant III: Both x and y are negative.
• Quadrant IV: x is positive and y is negative.

This plane allows us to graphically represent equations and inequalities, making it easier to visualize mathematical concepts. The clear layout helps in analyzing relationships between points and geometric shapes, making it a useful tool in solving exercises such as finding midpoints. Through this system, complex geometric problems are simplified into manageable algebraic equations, enhancing clarity and efficiency.

Mathematical Calculations

Mathematical calculations are fundamental to solving problems in coordinate geometry. They allow us to manipulate formulas to find solutions like distances, slopes, and in this instance, midpoints. The midpoint formula is an essential tool here, defined by:\[\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\]This formula calculates the average of the x-coordinates and the y-coordinates of two points. It tells you the exact point that is midway between the two.

• For example, calculating the midpoint between points \((2,7)\) and \((4,3)\):
  First, find the sum of the x-coordinates: \(2 + 4 = 6\).
• Then, divide by 2 to find the average: \(\frac{6}{2} = 3\).
• Repeat the same for the y-coordinates: \(7 + 3 = 10\), then \(\frac{10}{2} = 5\).
• The midpoint is therefore \((3,5)\).

This step-by-step calculation highlights how arithmetic can be applied effectively to find geometric solutions. Mathematical calculations in coordinate geometry not only provide the methods to solve problems but also reinforce the connection between algebra and geometry, ensuring deeper understanding and competency.