Problem 14 - Algebra 1: Concepts and Skills

Question

Find the distance between the two points. Round your solution to the nearest hundredth if necessary. $$ (-3,1),(2,6) $$

Step-by-Step Solution

Answer

The distance between the points (-3,1) and (2,6) is approximately 7.07.

Step 1: Identify the coordinates

The points are (-3,1) and (2,6). This means that \((x_1, y_1) = (-3, 1)\) and \((x_2, y_2) = (2, 6)\).

Step 2: Plug the coordinates into the formula

Substitute the values into the distance formula: \(d = \sqrt{{(x_2 - x_1)^2 + (y_2 - y_1)^2}}\), we get \(d = \sqrt{{(2 - (-3))^2 + (6 - 1)^2}}\).

Step 3: Solve the equation

Solve the equation to find the distance: \(d = \sqrt{{(5)^2 + (5)^2}} = \sqrt{{25+25}} = \sqrt{50}\).

Step 4: Simplify the number

Simplify the value of \(d\) and if necessary round it to the nearest hundredth. Thus, \(d = \sqrt{50} \approx 7.07\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a fascinating branch of mathematics that allows us to represent and analyze geometric figures, such as points, lines, and shapes, using coordinates. In this method, each point in the plane is determined by an ordered pair of numbers.
These numbers are usually written as \(x, y\), where \x\ represents the horizontal position (along the x-axis) and \y\ represents the vertical position (along the y-axis).

By using coordinate geometry, we can take complex geometric problems and convert them into simpler algebraic expressions. This provides a powerful tool for solving problems that involve geometric figures in mathematics, physics, engineering, and computer sciences.

• Each point in a 2D system is defined by its coordinates \(x\) and \(y\).
• Simplifies geometric problems through algebraic manipulation.
• Useful in a wide range of scientific and practical applications.

Distance Calculation

The distance formula is a critical tool within coordinate geometry used to determine the distance between two points. This formula is derived from the Pythagorean theorem, which relates the sides of a right triangle. Given two points \( (x_1, y_1) \) and \( (x_2, y_2) \), the formula to find the distance \(d\) between them is:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Here, you start by subtracting the coordinates of the first point from the corresponding coordinates of the second point. Then, each difference is squared. The results are added together and the square root of this sum gives the distance.

• The formula reflects a real-world geometric distance.
• Involves basic operations: subtraction, squaring, addition, and square root.
• Provides precise measurement between points on a plane.

Mathematics Problem Solving

Mathematics problem solving involves understanding the problem, devising a plan, carrying out the plan, and then evaluating the solution for accuracy. The exercise provided is an example of applying systematic steps to solve a distance calculation problem.

Breaking down this problem:

• Identify the problem: Determine what is being asked. Here, it's finding the distance between two points.
• Devise a plan: Use the distance formula from coordinate geometry.
• Carry out the plan: Accurately plug in numbers and simplify the expression.
• Check the work: Ensure calculations are correct, rounding as instructed, typically to the nearest hundredth in real-world applications.

This structured approach enhances understanding and helps to attack problems in a logical and organized manner, ensuring accuracy and understanding beyond rote calculation.