Question

In Exercises 27 and \(28,\) find the coordinates of the midpoint of the line segment joining the points. (5,-9,7),(-2,3,3)

Short answer

The coordinates of the midpoint of the line segment joining the points (5,-9,7) and (-2,3,3) are (1.5, -3, 5).

Step-by-step solution

Identify the coordinates of the given points

The first point is (x1,y1,z1) = (5,-9,7) and the second point is (x2,y2,z2) = (-2,3,3).

Apply the midpoint formula

Apply the formula for the midpoint of a line segment to find the x, y, and z coordinates of the midpoint.\n M = ((x1+x2)/2 , (y1+y2)/2 , (z1+z2)/2)\n Substitute the given coordinates in the formula:\n M = ((5-2)/2 , (-9+3)/2 , (7+3)/2)

Calculate the coordinates of the midpoint

Perform the calculations:\n M = ((5-2)/2 , (-9+3)/2 , (7+3)/2) = (3/2, -6/2, 10/2)\n Simplify the fractions to get the midpoint coordinates:\n M = (1.5, -3, 5)

Key concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This method contrasts with synthetic geometry, where no use of numbers to define points or lines is made. In coordinate geometry, points are placed on a grid called a coordinate system. The most common system is the Cartesian coordinate system, which consists of perpendicular axes, typically labeled as the x-axis (horizontal), y-axis (vertical), and, in the case of three dimensions, a z-axis that indicates depth.

Each point in this system is defined by an ordered pair or triplet of numbers known as coordinates. For instance, a point in a 2D space is represented by (x, y), while a three-dimensional point is represented by (x, y, z). These coordinates allow for the precise description of geometrical shapes, positions, and sizes within this defined space, enabling the calculation of distances, midpoints, and slopes, among others.

Line Segment

A line segment is part of a line bounded by two distinct end points, and contains every point on the line between its endpoints. Unlike a line, which extends infinitely in both directions, a line segment has a measurable length. Students often encounter line segments as the sides of polygons or the edges of solids in geometry.

The length of a line segment is its most fundamental property and can be calculated using the distance formula based on the coordinates of its endpoints. In three-dimensional space, this length is determined by considering the x, y, and z dimensions, requiring a version of the distance formula that incorporates all three axes. Beyond its length, knowledge about the midpoint of a line segment can be particularly useful when constructing geometrical shapes, optimizing paths, and solving problems involving averages.

3D Space

The concept of 3D space extends the two-dimensional plane by adding a third dimension, usually denoted as z. This addition allows for a more comprehensive representation of the physical world where depth becomes significant. In 3D coordinate geometry, points are identified by three coordinates (x, y, z), creating a three-dimensional grid structure called the Cartesian space.

In this space, geometry becomes more complex and intriguing, as it includes the study of objects such as spheres, cylinders, and polyhedra. Calculations that are simple in 2D, such as finding distances, areas, and midpoints, now require consideration of the z coordinate. Applying the midpoint formula in 3D space involves calculating the average of the x, y, and z coordinates of two points, allowing us to pinpoint the center of a line segment in three dimensions, as illustrated in the given exercise solution.