Problem 10 - Prealgebra

Question

Find the distance between each pair of points. Round to the nearest tenth, if necessary. $$M(0,0), N(-7,-8)$$

Step-by-Step Solution

Answer

The distance between points M and N is 10.6 units.

Step 1: Understand the Distance Formula

The distance formula, used to find the distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \), is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] where \(d\) is the distance.

Step 2: Identify Point Coordinates

Identify the coordinates of the points M and N. For point M \( (x_1, y_1) \), the coordinates are (0, 0). For point N \( (x_2, y_2) \), the coordinates are (-7, -8).

Step 3: Substitute Coordinates into the Formula

Substitute the coordinates into the distance formula: \[ d = \sqrt{((-7) - 0)^2 + ((-8) - 0)^2} \] which simplifies to \[ d = \sqrt{(-7)^2 + (-8)^2} \].

Step 4: Calculate Squared Differences

Calculate the squared differences: \((-7)^2 = 49\) and \((-8)^2 = 64\).

Step 5: Sum the Squared Differences

Add the squared differences: \[ 49 + 64 = 113 \]

Step 6: Calculate the Square Root

Now, calculate the square root of 113 to find the distance: \[ d = \sqrt{113} \] which approximately equals 10.6301.

Step 7: Round to the Nearest Tenth

Round the result to the nearest tenth: 10.6301 rounds to 10.6.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry to find solutions for geometric problems using a coordinate system. In this system, the position of any point can be defined using a pair of numbers called coordinates. The coordinates are typically written as

• The first number: known as the x-coordinate, indicates the position along the horizontal axis.
• The second number: known as the y-coordinate, indicates the position along the vertical axis.

Coordinate geometry allows us to use formulas, like the distance formula or the midpoint formula, to discover relationships and measurements between points more systematically. In the exercise, you utilize the distance formula to determine the length of the segment between points M (0, 0) and N (-7, -8). This powerful application of coordinate geometry simplifies problem-solving in various real-world and mathematical contexts.

Rounding Numbers

Rounding numbers is an essential skill in mathematics, particularly useful for simplifying complex numbers for easier comprehension and communication. When you round a number, you reduce the number of digits, keeping its value close but often easier to work with.
In the example of calculating the distance between two points, after applying the distance formula, the result was approximately 10.6301.
You should round this to the nearest tenth, which means finding the first decimal place.

• If the next digit is 5 or higher, you round up.
• If it is less than 5, you round down.

For this exercise, 10.6301 rounds to 10.6. Rounding helps make numbers more manageable and can be especially handy in real-world situations involving measurements, money, or estimates.

Square Roots

In mathematics, a square root is a value which, when multiplied by itself, gives the original number. The square root of a number is denoted with a radical symbol (\( \sqrt{} \)) and often calculated for various mathematical computations, such as the distance formula.
Understanding square roots is crucial for simplifying complex expressions and solving equations.

• For our exercise, you need to calculate the square root of 113;
• Because it isn't a perfect square, you estimate the square root to be approximately 10.6301;

Mastering square roots enables solving many problems in geometry and algebra by recognizing numerical patterns and relationships.

Prealgebra Concepts

Prealgebra lays the groundwork for future algebraic learning, covering fundamental ideas and skills needed to understand and perform basic mathematical operations. It focuses on essential mathematical concepts such as integers, fractions, decimals, and fundamental geometry. Prealgebra explores methods for:

• Understanding operations like addition, subtraction, multiplication, and division;
• Grasping the relationship between numbers and their operations;
• Introducing simple formulas like the distance formula for finding unknown values.

In this task, you used core prealgebra concepts by learning how to use a simple formula, substitute numerical values, and perform arithmetic operations. This practice is a building block for more advanced math courses in the future.