Question

Find the midpoint of the line segment joining the points \(P_{1}\) and \(P_{2}\). $$ P_{1}=(-1,4) ; \quad P_{2}=(8,0) $$

Short answer

Midpoint = \( (3.5, 2) \)

Step-by-step solution

Understand the Midpoint Formula

The midpoint of a line segment joining two points \(P_1 (x_1, y_1)\) and \(P_2 (x_2, y_2)\) can be found using the formula: \(\text{Midpoint} = \bigg( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \bigg)\).

Identify the Coordinates

Identify the coordinates of the given points: \(P_1 = (-1, 4)\) and \(P_2 = (8, 0)\). Here, \(x_1 = -1\), \(y_1 = 4\), \(x_2 = 8\), and \(y_2 = 0\).

Apply the Midpoint Formula

Substitute the coordinates into the midpoint formula: \( \bigg( \frac{-1 + 8}{2}, \frac{4 + 0}{2} \bigg) \).

Perform the Calculations

Calculate the midpoint: \( \bigg( \frac{7}{2}, \frac{4}{2} \bigg) = \bigg(3.5, 2\bigg) \).

Key concepts

midpoint formula

In coordinate geometry, one of the essential concepts is finding the midpoint of a line segment. The midpoint is essentially the average of the x-coordinates and y-coordinates of the two endpoints. This is helpful in various fields like physics, engineering, and computer graphics. The formula to find the midpoint between two points \( P_1 (x_1, y_1) \) and \( P_2 (x_2, y_2) \) is: \[ \text{Midpoint} = \bigg( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \bigg) \]
To use this formula, simply:

• Add the x-coordinates of both points together and divide by 2.
• Add the y-coordinates of both points together and divide by 2.

This results in the x and y coordinates of the midpoint. For example, for points \(P_1 = (-1, 4)\) and \(P_2 = (8, 0)\), the midpoint computation looks like this:
\[ \bigg( \frac{-1 + 8}{2}, \frac{4 + 0}{2} \bigg) = \bigg( \frac{7}{2}, \frac{4}{2} \bigg) = (3.5, 2) \]

coordinate geometry

Coordinate geometry, also called analytic geometry, allows us to study geometry using a coordinate system. This field combines algebra and geometry and provides a connection between algebraic equations and geometric figures.

• In coordinate geometry, we describe the position of points using ordered pairs (x, y).
• We can find distances, midpoints, and slopes of line segments using formulas.

Employing coordinate geometry helps to simplify complex geometric problems.
For instance, when asked to find the midpoint of a line segment, the problem is vastly simplified by employing the midpoint formula within the coordinate plane. This means you do not need to draw and measure on the graph but just rely on arithmetic operations to get the precise value.

line segment

A line segment is a portion of a line that is bounded by two distinct end points. Line segments are different from lines because they have a definite start and end, whereas lines extend infinitely in both directions.
The properties and measures of line segments can be calculated using basic geometrical and algebraic methods. For example, you can:

• Find the length of the line segment using the distance formula.
• Determine the midpoint of the segment for dividing it into two equal parts.

In the example provided, segment \( \text{joining points} P_1 = (-1, 4) \text{ and } P_2 = (8, 0) \) was analyzed. Here, computing the midpoint helps to understand the point that lies exactly halfway between these two ends. This is a common task in various practical applications and in solving more complex geometrical problems.