Problem 21 - Algebra 2

Question

Find the distance between each pair of points with the given coordinates. $$ (2.3,-1.2),(-4.5,3.7) $$

Step-by-Step Solution

Answer

The distance is approximately 8.38 units.

Step 1: Understand the Distance Formula

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

Step 2: Identify Coordinates

Identify the x and y coordinates from the given points. Here, \(x_1 = 2.3\), \(y_1 = -1.2\), \(x_2 = -4.5\), and \(y_2 = 3.7\).

Step 3: Calculate Differences

Calculate the differences between the x coordinates and the y coordinates:\[ x_2 - x_1 = -4.5 - 2.3 = -6.8 \]\[ y_2 - y_1 = 3.7 - (-1.2) = 4.9 \].

Step 4: Square the Differences

Square the differences calculated in the previous step:\[ (-6.8)^2 = 46.24 \]\[ (4.9)^2 = 24.01 \].

Step 5: Sum the Squares

Add the squares of the differences:\[ 46.24 + 24.01 = 70.25 \].

Step 6: Apply Square Root

Find the square root of the sum to determine the distance:\[ d = \sqrt{70.25} \].

Step 7: Calculate Distance

Compute the final value:\[ d \approx 8.38 \]. The distance between the points is approximately 8.38 units.

Key Concepts

Coordinate Geometry

Coordinate Geometry is a crucial mathematical field that connects algebra and geometry. It allows us to precisely locate points on a plane using ordered pairs known as coordinates. Every point in a two-dimensional plane can be identified by combining its x-coordinate and y-coordinate. These values essentially describe how far the point is from the origin along the horizontal (x-axis) and vertical (y-axis).

The use of coordinate geometry simplifies the analysis of geometric shapes, making it possible to calculate distances, midpoints, and other geometric properties using algebraic methods. In the exercise, the given points are (2.3, -1.2) and (-4.5, 3.7). Here, each pair's first number represents the x-coordinate, and the second represents the y-coordinate.

This representation is fundamental for calculating the distance between these points, which leads us to our next core concept: Distance Calculation.

Distance Calculation

Distance Calculation in coordinate geometry involves finding how far apart two points are on a plane. To achieve this, we use the Distance Formula. This formula, derived from the Pythagorean Theorem, offers a consistent method for calculating geometric distances using coordinates.

The formula is:

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

This formula calculates the direct line (distance from one point to another, resembling the hypotenuse of a right triangle created by these points with respect to the axes.

For the points provided in the exercise, those calculations involve:

• Subtraction of coordinates: \((x_2-x_1) = (-4.5 - 2.3) = -6.8\) and \((y_2-y_1) = (3.7 - (-1.2)) = 4.9\).
• Then, square the results to guarantee positive values: \((-6.8)^2 = 46.24\) and \((4.9)^2 = 24.01\).
• Add both squares to find the sum: \(46.24 + 24.01 = 70.25\).

This sum now requires further processing, transitioning us into the concept of Square Root Calculation.

Square Root Calculation

Square Root Calculation is an integral mathematical operation used to find a number which, when multiplied by itself, gives the original number. In the context of the Distance Formula, applying the square root is the final step to retrieve the actual distance value.

When you square a negative difference in coordinates (as seen in our exercise), it removes any directional component, leaving only the magnitude. This ensures that the distance is always a positive number.

Starting with the summed squares calculation from earlier:

• The sum was \(70.25\).

We find the square root of this sum to obtain the distance:

• \(d = \sqrt{70.25} \approx 8.38\)

This result is the direct spatial distance between the two points on the plane, showcasing the elegance and utility of coordinate geometry in simplifying otherwise complex geometric problems.