Problem 19 - Calculus an Applied Approach

Question

Find the coordinates of the midpoint of the line segment joining the two points. $$ (-5,-2,5),(6,3,-7) $$

Step-by-Step Solution

Answer

The coordinates of the midpoint of the line segment joining the two points (-5,-2,5) and (6,3,-7) are (1/2, 1/2, -1).

Step 1: Identify the Coordinates of the Given Points

The coordinates of the first point, (Point A), are (-5, -2, 5). The coordinates of the second point, (Point B), are (6, 3, -7).

Step 2: Apply the Formula of the Midpoint for All Three Coordinates

Apply the formula of the midpoint: \( ((x1+x2)/2, (y1+y2)/2, (z1+z2)/2) \) for all three coordinates X, Y and Z. This means for X it would be \(((-5+6)/2)\), for Y \(((-2+3)/2)\), and for Z \(( (5+(-7))/2)\).

Step 3: Simplify the Results

After calculating the previous expressions, the X coordinate is \((1/2)\), Y coordinate is \((1/2)\) and the Z coordinate is \((-1)\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This system often involves the placement of geometric figures in a plane using numbers, known as coordinates. Each point in the plane is represented by an ordered pair \(x, y\). In three-dimensional geometry, this is extended to an ordered triple \(x, y, z\). By using coordinate geometry, we can conduct algebraic representations and calculations involving geometric figures.

This approach enables us to solve problems related to distances and midpoints, like the one given. The midpoint formula specifically allows us to find the exact middle point of a line segment. Applying coordinate geometry to solve such problems offers an efficient and reliable method to explore spatial relations and geometric transformations.

Three-Dimensional Geometry

In three-dimensional geometry, we expand the principles of two-dimensional geometry into an extra dimension. This is widely applicable in real-world situations. Coordinates in three-dimensional space are written as \(x, y, z\), adding a third number that conveys depth.

In the context of finding the midpoint of a line segment in 3D space, each coordinate of the midpoint is the average of the corresponding coordinates of the endpoints of the line segment. The formula applied is:

• Midpoint for X = \(\frac{x_1 + x_2}{2}\)
• Midpoint for Y = \(\frac{y_1 + y_2}{2}\)
• Midpoint for Z = \(\frac{z_1 + z_2}{2}\)

This method provides the position of the midpoint along the X, Y, and Z axes. For example, given points \((-5, -2, 5)\) and \(6, 3, -7\), calculating the midpoint along each axis leads to the results \(0.5, 0.5, -1\). The ability to manipulate and understand 3D geometric concepts is crucial for many scientific and engineering tasks.

Line Segments

A line segment is the shortest path connecting two points. It consists of two endpoints and all the points that lie between them. In the context of coordinate geometry, line segments are often analyzed using their endpoints' coordinates.

To find a midpoint of a line segment, we take each coordinate of the endpoints and calculate their average. This gives us the coordinates of the midpoint, which evenly divides the segment into two equal halves.

• In a two-dimensional plane, if endpoints are \(A(x_1, y_1)\) and \(B(x_2, y_2)\), the midpoint, \(M\), has coordinates \((\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})\).
• Similarly, in three-dimensional space, the formula \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right)\) is used.

A proper understanding of line segments and their midpoints is key to mastering more complex geometric concepts. Recognizing how they are calculated helps in both theoretical and applied aspects of mathematics.