Problem 5 - Algebra 2

Question

Find the distance between each pair of points with the given coordinates. $$ (2,-4),(10,-10) $$

Step-by-Step Solution

Answer

The distance between the points is 10 units.

Step 1: Identify the given points

The problem gives us two points with coordinates: Point A at \((2, -4)\) and Point B at \((10, -10)\). We'll use these coordinates to calculate the distance between the points.

Step 2: Recall the distance formula

To calculate the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the distance formula: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\].

Step 3: Substitute the coordinates into the formula

Substitute \((2, -4)\) for \((x_1, y_1)\) and \((10, -10)\) for \((x_2, y_2)\). The formula becomes: \[d = \sqrt{(10 - 2)^2 + (-10 - (-4))^2}\].

Step 4: Calculate the differences

Calculate the differences: For the x-coordinates: \(10 - 2 = 8\) For the y-coordinates: \(-10 - (-4) = -10 + 4 = -6\).

Step 5: Square the differences

Now, square the differences obtained in the previous step:\(8^2 = 64\)\((-6)^2 = 36\).

Step 6: Add the squared differences

Add the squared differences: \[64 + 36 = 100\].

Step 7: Take the square root to find the distance

Finally, take the square root of the sum to find the distance: \[d = \sqrt{100} = 10\].

Key Concepts

Distance Between Points

The distance between two points in a plane is a measure of how far apart the points are. This concept is fundamental in coordinate geometry and is widely used in various applications across mathematics and science.
To find the distance between points, we use the distance formula. The distance formula is derived from the Pythagorean theorem, which links algebra with geometry to find distances in a plane.
The formula is: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
Here, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points. This formula helps us find the length of the hypotenuse of a right-angled triangle, where the two other sides are the differences in the x and y coordinates.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This branch of mathematics uses algebraic equations to represent geometric figures and spatial relationships.

Here are some of the core concepts:

• Coordinate Plane: A plane with a horizontal axis (x-axis) and a vertical axis (y-axis) used to locate points using pairs of numbers known as coordinates.
• Points: Defined by an ordered pair of numbers, \((x, y)\), representing their positions on the coordinate plane.
• Distance Formula: Utilized to calculate the distance between two points by leveraging their coordinates.

Coordinate geometry is pivotal in bridging algebra and geometry, allowing problems in one domain to be solved using methods from the other.

Mathematics Education

Mathematics education encompasses teaching and learning practices aimed at developing students' mathematical understanding and problem-solving skills. A clear understanding of mathematical concepts, like the Distance Formula, is crucial as it forms the foundation for higher-level mathematics concepts.

Strategies to enhance learning include:

• Hands-on Activities: Engaging students in interactive tasks such as plotting points on graph paper can solidify understanding.
• Visual Aids: Diagrams and interactive geometric tools can help students visualize mathematical concepts.
• Practice: Providing ample exercises for students to practice using formulas and solving problems.

Emphasizing connections between algebra and geometry through exercises such as calculating distances between points nurtures a well-rounded mathematical proficiency.

Problem Solving in Algebra

Problem solving in algebra involves translating real-world situations or geometric contexts into mathematical statements or equations. The goal is to solve these equations to find unknowns or derive certain values. Calculating the distance between points is a typical algebra problem-solving task.

• Identifying Given Variables: Start by identifying the coordinates of the points involved.
• Applying the Correct Formula: Use the distance formula by substituting the known values.
• Performing Calculations Carefully: The steps include calculating differences, squaring them, summing the squares, and finally taking the square root.

Through these steps, students practice logical reasoning, systematic planning, and diligent computation, all fundamental skills in algebraic problem-solving.