Problem 4 - College Algebra Essentials

Question

find the distance between each pair of points. If necessary, round answers to two decimals places. $$ (2,-3) \text { and }(-1,5) $$

Step-by-Step Solution

Answer

The distance between the points (2,-3) and (-1,5) is approximately 8.54 units when rounded to two decimal places.

Step 1: Identify Points and Formula

Identify the points given. The points are (2,-3) and (-1,5). The formula for calculating the distance between them is \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Now, it's time to substitute our points into this formula.

Step 2: Substitution

Let's substitute the values into the distance formula. We have \( x_1 = 2, y_1 = -3, x_2 = -1, y_2 = 5 \). So, the distance \( D \) will be: \( D = \sqrt{(-1 - 2)^2 + (5 - (-3))^2} \).

Step 3: Solve the Expression

Now, let's solve the equation. The equation becomes \( D = \sqrt{(-3)^2 + (8)^2} = \sqrt{9 + 64} = \sqrt{73} \).

Step 4: Round the Answer

Finally, round your answer to two decimal places if necessary. The approximated value of \( \sqrt{73} \) to two decimal places is 8.54.

Key Concepts

Coordinate Plane

To understand distance calculation in algebra, it is essential to start with the fundamental concept of the coordinate plane. The coordinate plane, also known as the Cartesian plane, is formed by two perpendicular number lines intersecting at a point called the origin. The horizontal line is called the x-axis, and the vertical line is the y-axis. Together, they divide the plane into four quadrants.

Each point on the plane is defined by a pair of numerical coordinates: the first number, or the x-coordinate, corresponds to the position on the x-axis, while the second number, or the y-coordinate, corresponds to the position on the y-axis. For instance, the point (2, -3) lies 2 units to the right of the origin on the x-axis and 3 units down on the y-axis.

Distance Calculation

Distance calculation on the coordinate plane is the process of determining the length of the segment connecting two points. It employs the distance formula derived from the Pythagorean theorem, which applies to a right-angled triangle formed by the points and their projections onto the axes.

To calculate the distance between two points, take the x-coordinates and y-coordinates of both points and plug them into the distance formula: \( D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Make sure to follow the order of operations: first, calculate the differences, square them, add the squares together, and then take the square root.

Pythagorean Theorem

The Pythagorean theorem is a cornerstone of geometry, particularly when dealing with right-angled triangles. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Expressing this theorem algebraically gives us \( c^2 = a^2 + b^2 \), where \( c \) is the hypotenuse and \( a \) and \( b \) are the other two sides.

Application in Distance Calculation

When working with the coordinate plane, this theorem helps calculate the distance between two points, as the line segment connecting the points acts as the hypotenuse. By projecting this line onto the x-axis and the y-axis, two sides of a right-angled triangle are formed, allowing the application of the Pythagorean theorem to find the distance.

Rounding Decimals

Rounding decimals is a numerical technique used to reduce the number of digits after the decimal point, making the number simpler and easier to work with, especially when an exact value is not necessary. To round to a specific decimal place, look at the digit to the right of the desired decimal place. If this digit is 5 or more, increase the digit in the desired place by one. If it's less than 5, leave the digit unchanged.

In our example, the value \( \sqrt{73} \) approximately equal to 8.544 rounded to two decimal places becomes 8.54. Rounding is essential for reporting a clear and concise answer in various fields, including mathematics, science, and finance.