Problem 18 - Precalculus

Question

find the midpoint of the line segment joining points \(P\) and \(Q\) (a) \(P(12,0)\) and \(Q(12,8)\) (b) \(P\left(\frac{3}{5},-\frac{2}{3}\right)\) and \(Q(0,0)\) (c) \(P(1, \pi)\) and \(Q(3,3 \pi)\)

Step-by-Step Solution

Answer

(a) \((12, 4)\), (b) \(\left(\frac{3}{10}, -\frac{1}{3}\right)\), (c) \((2, 2\pi)\)."

Step 1: Understand the Formula

To find the midpoint of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\), we use the midpoint formula: \[M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]This formula calculates the averages of the x-coordinates and the y-coordinates of the endpoints.

Step 2: Apply Formula to Part (a)

For points \(P(12,0)\) and \(Q(12,8)\), apply the formula:\[M = \left( \frac{12 + 12}{2}, \frac{0 + 8}{2} \right)\]Calculate:\[M = \left( \frac{24}{2}, \frac{8}{2} \right) = (12, 4)\]The midpoint is \((12, 4)\).

Step 3: Apply Formula to Part (b)

For points \(P\left(\frac{3}{5}, -\frac{2}{3}\right)\) and \(Q(0,0)\), apply the formula:\[M = \left( \frac{\frac{3}{5} + 0}{2}, \frac{-\frac{2}{3} + 0}{2} \right)\]Calculate:\[M = \left( \frac{3}{10}, -\frac{1}{3} \right)\]The midpoint is \(\left(\frac{3}{10}, -\frac{1}{3}\right)\).

Step 4: Apply Formula to Part (c)

For points \(P(1, \pi)\) and \(Q(3, 3\pi)\), apply the formula:\[M = \left( \frac{1 + 3}{2}, \frac{\pi + 3\pi}{2} \right)\]Calculate:\[M = \left( 2, 2\pi \right)\]The midpoint is \((2, 2\pi)\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic methods to solve geometric problems. It acts as a bridge between algebra and geometry through graphs and equations representing geometric figures. By assigning coordinates to geometric figures, coordinate geometry helps us to easily calculate distances, midpoints, and other properties of these figures in a systematic way.
In a coordinate system, every point is defined by an ordered pair \(x, y\). These pairs correspond to positions on a plane where \(x\) is the horizontal position and \(y\) is the vertical position. By manipulating these coordinates using algebraic equations, we can gain insights into the geometric properties of the figures they define.
In exercises like finding the midpoint, coordinate geometry simplifies the process significantly. Instead of requiring intricate geometric constructions, it provides a straightforward algebraic formula to find a solution.

Line Segment

A line segment in geometry is part of a line bounded by two distinct endpoints. Unlike a line, a line segment does not extend indefinitely. It has a fixed length determined by its endpoints. Understanding the properties of line segments helps in solving various problems, especially in coordinate geometry.
For example, when given the endpoints \(P\) and \(Q\), one can easily determine the location of a midpoint along the segment by calculating the average of the coordinates. Line segments are fundamental elements in geometry as they often represent simple, direct connections between points. Their length can be measured as the distance between the endpoints, and their location as the average of the endpoints' coordinates.

• Each line segment can be represented uniquely by its two endpoints.
• A line segment has two directions, meaning it could extend from \(P\) to \(Q\) or from \(Q\) to \(P\).

Calculating characteristics such as their midpoint or length helps in advancing the understanding of the geometric figures they part.

Averaging Coordinates

The process of finding a midpoint involves averaging coordinates, an essential concept in coordinate geometry. When tasked with finding the midpoint of a line segment, you do this by averaging the x-coordinates, and separately averaging the y-coordinates, of the endpoints.
The midpoint formula \(M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\) is a direct application of averaging coordinates. Here's a step-by-step look at this:

• Identify the coordinates of the two endpoints.
• Calculate the average of the x-coordinates: \((x_1 + x_2) / 2\).
• Calculate the average of the y-coordinates: \((y_1 + y_2) / 2\).
• Combine these averages to find the midpoint \((M)\).

Understanding this averaging concept is crucial as it simplifies the process of dividing the segment into two equal parts, thus providing the precise central point or location on the segment.