Problem 17 - Essential Calculus Early Transcendentals

Question

(a) Prove that the midpoint of the line segment from \(P_{1}\left(x_{1}, y_{1}, z_{1}\right)\) to \(P_{2}\left(x_{2}, y_{2}, z_{2}\right)\) is $$\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}, \frac{z_{1}+z_{2}}{2}\right)$$ (b) Find the lengths of the medians of the triangle with vertices \(A(1,2,3), B(-2,0,5),\) and \(C(4,1,5) .\)

Step-by-Step Solution

Answer

(1) Midpoint: \( \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2} \right) \); (2) Medians' lengths: \(2.5, \sqrt{23.5}, \sqrt{21.25}\).

Step 1: Understanding Midpoint Formula

The midpoint of a line segment with endpoints \(P_1(x_1, y_1, z_1)\) and \(P_2(x_2, y_2, z_2)\) is the point that lies exactly halfway between those two endpoints. This point's coordinates are calculated as the average of the corresponding coordinates of the endpoints.

Step 2: Deriving the Midpoint Formula

To find the midpoint, average the \(x\), \(y\), and \(z\) coordinates separately. The formula becomes: \[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right). \] This represents the midpoint, as it is equidistant from both \(P_1\) and \(P_2\).

Step 3: Calculating Medians' Lengths

For triangle vertices \(A(1,2,3)\), \(B(-2,0,5)\), and \(C(4,1,5)\), a median is a line segment joining a vertex to the midpoint of the opposite side. First, find the midpoint of each side using the midpoint formula:- Midpoint of \(BC\): \(M_{BC} = \left( \frac{-2+4}{2}, \frac{0+1}{2}, \frac{5+5}{2} \right) = (1, 0.5, 5)\)- Midpoint of \(AC\): \(M_{AC} = \left( \frac{1+4}{2}, \frac{2+1}{2}, \frac{3+5}{2} \right) = (2.5, 1.5, 4)\)- Midpoint of \(AB\): \(M_{AB} = \left( \frac{1-2}{2}, \frac{2+0}{2}, \frac{3+5}{2} \right) = (-0.5, 1, 4)\)

Step 4: Calculating Length of each Median

Use the distance formula to calculate the lengths of each median from the vertex to its opposite midpoint:- Length of \(AM_{BC}\): \[ \sqrt{(1-1)^2 + (2-0.5)^2 + (3-5)^2} = \sqrt{0 + 2.25 + 4} = \sqrt{6.25} = 2.5 \]- Length of \(BM_{AC}\): \[ \sqrt{(-2-2.5)^2 + (0-1.5)^2 + (5-4)^2} = \sqrt{(-4.5)^2 + (-1.5)^2 + 1} \] \[ = \sqrt{20.25 + 2.25 + 1} = \sqrt{23.5} \]- Length of \(CM_{AB}\): \[ \sqrt{(4+0.5)^2 + (1-1)^2 + (5-4)^2} = \sqrt{(4.5)^2 + 0 + 1} = \sqrt{20.25 + 1} \] \[ = \sqrt{21.25} \]

Step 5: Conclude the Calculation

The lengths of the medians are:1. \(AM_{BC} = 2.5\)2. \(BM_{AC} = \sqrt{23.5}\)3. \(CM_{AB} = \sqrt{21.25}\).

Key Concepts

Distance Formula

The distance formula is essential for finding the length between two points in a coordinate plane or even in 3D space. If you have two points, say \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \),the distance between them can be found using the following formula:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\] This equation is an extension of the Pythagorean theorem into three dimensions.The 3D distance formula allows you to measure the straight-line distance between any two points in a three-dimensional coordinate system. It is extremely useful in applications like calculating the length of medians in triangles, as you might find in geometric problems. By substituting the values of the coordinates into this formula, you can easily compute distances without having to draw or visualize the distance in space.

Median of a Triangle

In geometry, a median is a crucial segment that connects a vertex of a triangle to the midpoint of its opposite side. Each triangle has three medians, one from each vertex. The medians are essential for various geometric constructions and calculations. They provide symmetric properties and balance within the triangle. To find a median, you first need to determine the midpoint of the opposite side using the midpoint formula. For example, if you are finding the median from vertex A to side BC , you calculate the midpoint of BC . Once you have the midpoint, you use the distance formula to determine the straight-line distance from the vertex to this midpoint, which gives you the length of the median. Medians have interesting properties, such as intersecting at a single point called the centroid, which divides each median into a 2:1 ratio. Exploring these segments helps in understanding the internal symmetry and balance of triangles.

3D Coordinate Geometry

3D coordinate geometry is an extension of 2D geometry into the three-dimensional space, allowing for the analysis of points, lines, and shapes beyond a flat surface.

• A point in 3D space is represented by three coordinates \((x, y, z)\), signifying its positional location in three mutually perpendicular directions.
• Line segments can exist between any two points, necessitating the use of the distance formula to measure their lengths in this space.

Coordinate geometry in three dimensions is fundamental for calculations involving planes, lines, and solids. This framework is particularly powerful as it allows for multidimensional analysis of geometric entities.The process involves extending techniques like the midpoint and distance formulas, learned in 2D geometry, into 3D. This allows one to solve more complex problems, such as those involving spatial relationships, angles between planes and lines, and determining the location of geometric centers like centroids in space. As you practice problems in this area, you begin to visualize and understand the characteristics and behaviors of shapes in a three-dimensional context.