Problem 42 - Algebra 1

Question

Find the midpoint between the two points \((5,1),(1,-5)\)

Step-by-Step Solution

Answer

The midpoint between the points (5,1) and (1,-5) is (3,-2).

Step 1: Identify the Coordinates

The first point has coordinates \( (x_1,y_1) = (5,1) \) and the second point has coordinates \( (x_2,y_2) = (1,-5) \).

Step 2: Apply Midpoint Formula

The formula for the midpoint is \( ( \frac{x_1+x_2}{2} , \frac{y_1+y_2}{2} ) \) . Plugging in the values from step 1: \( ( \frac{5+1}{2} , \frac{1+(-5)}{2} ) \).

Step 3: Simplify

Simplifying gives the midpoint as \( (3,-2) \).

Key Concepts

Coordinate Geometry

Coordinate geometry is a branch of mathematics that connects algebra and geometry. It allows us to use algebraic equations to solve geometric problems.
In coordinate geometry, each point is defined by a pair of numerical coordinates, typically written as \(x, y\). These numbers represent the location of the point on the Cartesian plane.

• The plane consists of two axes: the horizontal axis (x-axis) and the vertical axis (y-axis).
• Where these two axes intersect is known as the origin, labeled as \(0, 0\).
• Any point on this plane is identified by its distance from the origin along the x-axis and y-axis.

Using coordinate geometry, we can solve problems such as finding midpoints of a line segment, understanding slopes, and recognizing the shapes of various curves. It brings a visual element to algebraic solutions, making math more accessible and intuitive.

Algebra Concepts

In the context of coordinate geometry, algebra provides the tools needed to perform calculations, such as finding a midpoint. Let's break down some key algebra concepts involved:

• Variables: Used to represent numbers. In our problem, \(x_1, y_1, x_2, \text{ and } y_2\) are variables that stand in for specific coordinates.
• Expressions: Combinations of variables and numbers connected by operations like addition and division, such as \(\frac{x_1 + x_2}{2}\).
• Formulas: Used to solve for specific pieces of information. For instance, the midpoint formula \(\left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right)\) helps us find the exact middle point between two coordinates.

The power of algebra in coordinate geometry lies in its ability to manipulate these variables and expressions to find precise solutions.

Graphing Points

Graphing points is the process of placing a point on the Cartesian plane using its coordinates. This is crucial for visualizing relationships and solutions such as the midpoint.
To graph a point:

• Start at the origin \(0, 0\).
• Move horizontally: The first number in the pair \(x\) tells you how far to move along the x-axis.
• Move vertically: The second number \(y\) indicates your movement along the y-axis.

For instance, to graph the point \(5,1\), start at the origin, move 5 units right, then 1 unit up. For the point \(1,-5\), start again at the origin, move 1 unit right, and then 5 units down.
Graphing these points allows you to see the line segment between them, and when you calculate the midpoint, you can easily verify this by plotting the midpoint on the graph. Seeing these points and midpoints on a graph can deepen understanding and make concepts more tangible.