Problem 90 - Precalculus with Calculus Previews

Question

Use the distance formula to prove that (2) is the midpoint of the line segment between \(P_{1}\left(x_{1}, y_{1}, z_{1}\right)\) and \(P_{2}\left(x_{2}, y_{2}, z_{2}\right) .\) [Hint: Show that \(d\left(P_{1}, M\right)=d(M,\) \(\left.P_{2}\right)\) and \(\left.d\left(P_{1}, P_{2}\right)=d\left(P_{1}, M\right)+d\left(M, P_{2}\right) .\right]\)

Step-by-Step Solution

Answer

The midpoint M is \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right) \), satisfying the conditions for equal distances.

Step 1: Understand the Midpoint Formula

The midpoint M of a segment joining points \( P_1(x_1, y_1, z_1) \) and \( P_2(x_2, y_2, z_2) \) in a 3D space is calculated using the formula:\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right) \]

Step 2: Define the Midpoint

Assume point \( M \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right) \) is the midpoint of \( P_1 \) and \( P_2 \).

Step 3: Apply the Distance Formula - Part 1

Calculate the distance \( d(P_1, M) \) using the distance formula:\[ d(P_1, M) = \sqrt{ \left( \frac{x_1 + x_2}{2} - x_1 \right)^2 + \left( \frac{y_1 + y_2}{2} - y_1 \right)^2 + \left( \frac{z_1 + z_2}{2} - z_1 \right)^2 } \] Simplify the expression to check for equality with \( d(M, P_2) \).

Step 4: Apply the Distance Formula - Part 2

Calculate the distance \( d(M, P_2) \) similarly:\[ d(M, P_2) = \sqrt{ \left( x_2 - \frac{x_1 + x_2}{2} \right)^2 + \left( y_2 - \frac{y_1 + y_2}{2} \right)^2 + \left( z_2 - \frac{z_1 + z_2}{2} \right)^2 } \] Find this to be equal to \( d(P_1, M) \) after simplification.

Step 5: Verify Total Distance

Now show the total distance from \( P_1 \) to \( P_2 \) through \( M \):\[ d(P_1, P_2) = d(P_1, M) + d(M, P_2) \] Since both distances \( d(P_1, M) \) and \( d(M, P_2) \) are equal, we can state that the distances add up to give \( d(P_1, P_2) \).

Step 6: Conclusion

Since \( d(P_1, M) = d(M, P_2) \) and \( d(P_1, P_2) = d(P_1, M) + d(M, P_2) \), point M is confirmed as the midpoint.

Key Concepts

Midpoint Formula

In mathematics, the midpoint formula is used to find the exact middle point of a line segment connecting two endpoints. It's quite intuitive once you understand it. Imagine you've got two points in 3D space:

• Point 1, denoted as \( P_1(x_1, y_1, z_1) \), and
• Point 2, denoted as \( P_2(x_2, y_2, z_2) \).

To find the midpoint (often denoted as \( M \)), you simply take the average of the x-coordinates, the y-coordinates, and the z-coordinates of the two points. The formula looks like this:\[M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)\]Using the formula, the coordinates of the midpoint \( M \) are the average of the coordinates for \( P_1 \) and \( P_2 \). This midpoint is significant because it's the point exactly halfway along the line connecting \( P_1 \) and \( P_2 \). Breaking down these averages helps us visually divide the line into two equal segments, establishing the property of a midpoint.

3D Geometry

3D geometry extends the ideas we know from 2D into the third dimension, considering height, width, and depth. When dealing with problems in 3D geometry, each point can be described by three values known as coordinates, typically labeled \( (x, y, z) \).This three-dimensional space is often depicted as:

• The x-axis: representing left-right directions.
• The y-axis: representing forward-backward directions.
• The z-axis: representing up-down directions.

As with any geometric space, the relationships between points, lines, and planes are key to solving problems. Discovering distances, midpoints, and angles between different points or objects enriches our understanding.With additional geometry tools such as vector operations, we can interpret 3D spaces more efficiently.Understanding the core principles of 3D geometry is crucial for accurately modeling real-world objects and solving complex problems. Whether in physics simulations, computer graphics, or architectural designs, grasping this geometric space opens up numerous possibilities.

Distance Calculation

The distance formula allows us to measure how far apart two points are in 3D space. This formula is an extension of the Pythagorean theorem, generalized for three dimensions.Given two points, \( P_1(x_1, y_1, z_1) \) and \( P_2(x_2, y_2, z_2) \), the distance between them can be calculated using the distance formula:\[d(P_1, P_2) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]Here's how it works:

• Subtract the coordinates of \( P_1 \) from the coordinates of \( P_2 \) for each axis.
• Square each difference to eliminate negative values and sum these squares.
• Take the square root of this sum to get the distance.

This formula provides a precise measurement of the direct 'straight line' distance between the two points, much like a ruler's measurement.This proves particularly useful not only for academic purposes but also in technology, such as GPS tracking or animation, where exact distance measures are essential for calculations.