Problem 9 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the distance between \((-2,3)\) and the midpoint of the segment joining \((-2,-2)\) and \((4,3)\).

Step-by-Step Solution

Answer

The distance is approximately 3.905.

Step 1: Calculate the Midpoint

To find the midpoint of the segment joining the points \((-2, -2)\) and \((4, 3)\), use the midpoint formula: \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). Substitute the given coordinates: \( M = \left( \frac{-2 + 4}{2}, \frac{-2 + 3}{2} \right) = (1, 0.5) \). The midpoint is \((1, 0.5)\).

Step 2: Use Distance Formula

Now, find the distance between the point \((-2, 3)\) and the midpoint \((1, 0.5)\). Use the distance formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Substitute these coordinates: \( d = \sqrt{(1 - (-2))^2 + (0.5 - 3)^2} \). Simplify to find: \( d = \sqrt{(3)^2 + (-2.5)^2} \).

Step 3: Simplify the Expression

Calculate the square values: \( 3^2 = 9 \) and \((-2.5)^2 = 6.25 \). Add these values: \( 9 + 6.25 = 15.25 \).

Step 4: Calculate the Distance

Finally, find the square root of 15.25: \( d = \sqrt{15.25} \approx 3.905 \). The distance between \((-2, 3)\) and the midpoint \((1, 0.5)\) is approximately 3.905.

Key Concepts

midpoint formula

The midpoint formula is a fundamental concept in coordinate geometry which helps you find the exact center point between two given points on a plane. It is like finding an average location where two points meet. When you have two points, say \( (x_1, y_1) \) and \( (x_2, y_2) \), the midpoint \( M \) is calculated using the formula: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]. This formula works because it simply averages the x-coordinates and y-coordinates of your points.
So if you have two points like \( (-2, -2) \) and \( (4, 3) \), you substitute them into the formula: \[ M = \left( \frac{-2 + 4}{2}, \frac{-2 + 3}{2} \right) = (1, 0.5) \].

• The result, \( (1, 0.5) \), perfectly divides the segment into two equal parts.
• Knowing the midpoint can be crucial for finding distances or making geometric constructions.

It's important because the midpoint reveals symmetry in shapes and figures on a coordinate plane, which can be applied in various real-world scenarios, like engineering and design.

coordinate geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This system uses a pair of numerical coordinates to uniquely specify points on a plane. These coordinates are measured along perpendicular axes, typically labeled as the x-axis and y-axis.
In this exercise, it's crucial because:

• It helps you accurately plot and visualize points such as \( (-2, 3) \), the midpoint \( (1, 0.5) \), and others on a graph.
• Using these coordinates, geometric properties and relationships can be derived or verified through algebraic methods.

This field bridges algebra and geometry, allowing the use of equations to represent geometric shapes. It also simplifies the process of proving geometric theorems because you rely on algebraic equations rather than purely relying on diagrams or visual representations.
Coordinate geometry is powerful in various applications including computer graphics, navigation systems, and even in complex scientific computations where spatial relationships matter.

Euclidean distance

Euclidean distance is a way of measuring straight-line distance between two points in a multi-dimensional space. It's like stretching a tape measure from one point to another to find out how far apart they are.
The formula for Euclidean distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]. This formula might look familiar, because it's a specific case of the Pythagorean theorem applied to the plane.
For our given points, \( (-2, 3) \) and \( (1, 0.5) \), we calculate the distance by substituting the values: \[ d = \sqrt{(1 - (-2))^2 + (0.5 - 3)^2} = \sqrt{3^2 + (-2.5)^2} = \sqrt{15.25} \approx 3.905 \].

• This distance represents the shortest path connecting the two points on a plane.
• Understanding Euclidean distance is essential in fields like data science for clustering, physics for motion, and any context where spatial relationships need to be analyzed.

In summary, the concept of Euclidean distance is foundational to analyzing and understanding spatial relationships between entities in both theoretical and practical applications.