Problem 86 - Algebra A Combined Function

Question

Find the distance between each pair of points. Give an exact distance and a three-decimal-place approximation. (9.6,2.5) and (-1.9,-3.7)

Step-by-Step Solution

Answer

The distance is \(\sqrt{170.69}\) or approximately 13.060.

Step 1: Identify the Points

We have two points: \((x_1, y_1) = (9.6, 2.5)\) and \((x_2, y_2) = (-1.9, -3.7)\).

Step 2: Apply the Distance Formula

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

Step 3: Calculate Differences

Calculate \(x_2 - x_1 = -1.9 - 9.6 = -11.5\) and \(y_2 - y_1 = -3.7 - 2.5 = -6.2\).

Step 4: Square the Differences

Square the differences: \((-11.5)^2 = 132.25\) and \((-6.2)^2 = 38.44\).

Step 5: Sum the Squares

Add the squared differences: \(132.25 + 38.44 = 170.69\).

Step 6: Take the Square Root

Find the square root of the sum: \(d = \sqrt{170.69} = 13.060\).

Step 7: Provide Exact Value and Approximation

The exact distance is \(\sqrt{170.69}\) and the approximate distance to three decimal places is 13.060.

Key Concepts

Coordinate Geometry

Coordinate geometry is a branch of mathematics that involves using coordinate points to understand and solve geometric problems. It allows us to explore the spatial relations and measurements of shapes on a plane. This form of geometry uses values like x and y to define the position of points.

In coordinate geometry, each point on a plane is represented by a pair of numerical coordinates. These coordinates describe its precise location relative to a reference axis, typically denoted as the x-axis (horizontal) and y-axis (vertical). By plotting these points, we can form lines, curves, and shapes, enabling us to perform calculations such as finding distances and angles.

A key feature of coordinate geometry is its ability to connect algebra with geometry, simplifying complex visual problems with straightforward arithmetic. This makes it much easier to handle problems involving measurements and relationships between different geometric figures.

Distance Calculation

Calculating the distance between two points on a plane involves using a specific formula known as the distance formula. This formula derives from the Pythagorean Theorem, which is a fundamental principle in geometry. By applying this theorem, we can find the straight-line distance between two points irrespective of their location in the coordinate plane.

To calculate the distance, we need to know the coordinates of the two points, say \((x_1, y_1)\) and \((x_2, y_2)\). The distance \(d\) is calculated using the formula:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]This formula gives the length of the "straight line" connecting the two points.

• Subtract the x-coordinates: \(x_2 - x_1\)
• Subtract the y-coordinates: \(y_2 - y_1\)
• Square both of these differences.
• Add these squared values.
• Finally, take the square root of the sum to find the distance.

By following these steps, you can precisely determine the distance, which can be useful in many real-world applications such as mapping, navigation, and resource management.

Square Roots

The square root is a mathematical function that retrieves the original number from its square. It is an essential concept in mathematics, particularly in the context of distance calculation. When calculating distances, the square root is used to derive the original geometric length from the sum of squared differences between coordinate points.

Let's consider the sum we obtain after calculating the squared differences: The example above results in a sum, such as 170.69. To find the actual distance, we calculate the square root of this sum: \(\sqrt{170.69} \). This process reverses the squaring operation, providing us with the linear distance between the points on a two-dimensional plane.

When dealing with square roots, it is vital to comprehend both the exact value and its decimal approximation. The exact value remains as a square root expression, while the decimal form is a calculated approximation for practical use. Understanding square roots helps with interpreting the outcome of a distance calculation correctly.

Points on a Plane

In coordinate geometry, points on a plane are the fundamental units used to represent locations. They are defined by their coordinates and are essential for any calculations we want to perform, such as measuring distances or identifying relationships between shapes.

Each point is represented by a pair of values \((x, y)\). For instance, a point labeled \((9.6, 2.5)\) means that this point is located 9.6 units along the x-axis and 2.5 units up the y-axis. Similarly, another point like \((-1.9, -3.7)\) would be located 1.9 units in the negative direction of the x-axis and 3.7 units downward on the y-axis.

Understanding points and their respective locations on a plane helps us make sense of geometric transformations and spatial arrangements. They enable easy visualization of how objects relate to each other within the two-dimensional space, thus serving as the groundwork for more complex geometric explorations.