(a) Prove that the midpoint of the line segment from P1(x1, y1, z1) to P2(x2, y2

step 1 To prove for our midpoints, A, we are saying that if the midpoints, midpoints of the line segmen

(a) Prove that the midpoint of the line segment from ...

\begin{equation} \begin{array}{l}{\text { (a) Prove that the midpoint of the line segment from }} \\ {P_{1}\left(x_{1}, y_{1}, z_{1}\right) \text { to } P_{2}\left(x_{2}, y_{2}, z_{2}\right) \text { is }}\end{array} \end{equation} \begin{equation} \left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}, \frac{z_{1}+z_{2}}{2}\right) \end{equation} \begin{equation} \begin{array}{l}{\text { (b) Find the lengths of the medians of the triangle with ver- }} \\ {\text { tices } A(1,2,3), B(-2,0,5), \text { and } C(4,1,5) \text { . ( A median of a }} \\ {\text { triangle is a line segment that joins a vertex to the midpoint }} \\ {\text { of the opposite side.) }}\end{array} \end{equation}

\begin{equation}
\begin{array}{l}{\text { (a) Prove that the midpoint of the line segment from }} \\ {P_{1}\left(x_{1}, y_{1}, z_{1}\right) \text { to } P_{2}\left(x_{2}, y_{2}, z_{2}\right) \text { is }}\end{array}
\end{equation}
\begin{equation}
\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}, \frac{z_{1}+z_{2}}{2}\right)
\end{equation}
\begin{equation}
\begin{array}{l}{\text { (b) Find the lengths of the medians of the triangle with ver- }} \\ {\text { tices } A(1,2,3), B(-2,0,5), \text { and } C(4,1,5) \text { . ( A median of a }} \\ {\text { triangle is a line segment that joins a vertex to the midpoint }} \\ {\text { of the opposite side.) }}\end{array}
\end{equation}

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Key Concepts

Midpoint Formula in Three-Dimensional Space

This concept involves finding the point that is exactly halfway between two points in 3D space by averaging the corresponding coordinates. The formula ( (x1+x2)/2, (y1+y2)/2, (z1+z2)/2 ) provides a systematic method to determine the midpoint of a line segment connecting any two points, which is fundamental in geometry, computer graphics, and spatial analysis.

Distance Formula in Three-Dimensional Space

The distance formula is used to calculate the length of the line segment connecting two points in 3D space. Based on the Pythagorean theorem, it computes the square root of the sum of the squares of the differences in corresponding coordinates. This formula is crucial for measuring lengths, verifying geometric properties, and solving problems involving spatial distances.

Median of a Triangle

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. This concept is important in geometry because the medians of a triangle intersect at a single point called the centroid, which has several applications in fields such as physics, engineering, and geometric design due to its balancing properties.

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