Question

Find the midpoint of the line segment joining the points \(P_{1}\) and \(P_{2}\). $$ P_{1}=(7,-5) ; \quad P_{2}=(9,1) $$

Short answer

(8, -2)

Step-by-step solution

Understand the midpoint formula

The midpoint of a line segment between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) can be determined by the formula: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] This formula calculates the average of the x-coordinates and the y-coordinates of the endpoints.

Substitute the given points into the formula

Using the points \( P_1 = (7, -5) \) and \( P_2 = (9, 1) \), substitute \( x_1 = 7, y_1 = -5 \) and \( x_2 = 9, y_2 = 1 \) into the midpoint formula: \[ M = \left( \frac{7 + 9}{2}, \frac{-5 + 1}{2} \right) \]

Calculate the midpoint

Perform the addition and division for both the x and y coordinates: \[ M = \left( \frac{7 + 9}{2}, \frac{-5 + 1}{2} \right) = \left( \frac{16}{2}, \frac{-4}{2} \right) = (8, -2) \] Thus, the midpoint is \( (8, -2) \).

Key concepts

Line Segment

A line segment is a part of a line that has two distinct endpoints. It's essentially the portion of a line that connects two points. In this problem, those points are given as \( P_{1}=(7,-5) \) and \( P_{2}=(9,1) \).

Line segments are important in geometry because they have a definite length which can be measured. They are the simplest figures that form the basis of more complex shapes.

To visualize this concept more clearly, imagine drawing a straight path from point \( P_{1} \) to point \( P_{2} \). That path is your line segment. Understanding this helps you see why we need to find a midpoint—it's the exact middle part of this path.

Midpoint

The midpoint of a line segment is the point that is exactly halfway between the segment’s endpoints. In simpler terms, it divides the line segment into two equal parts.
The formula for finding the midpoint \( M \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Let's break down this formula:

• \( \frac{x_1 + x_2}{2} \) is the average of the x-coordinates of the endpoints.
• \( \frac{y_1 + y_2}{2} \) is the average of the y-coordinates of the endpoints.

By calculating these averages, we find the coordinates of the midpoint. For our points \( P_1 = (7, -5) \) and \( P_2 = (9, 1) \), the midpoint would be:

\[ M = \left( \frac{7 + 9}{2}, \frac{-5 + 1}{2} \right) = (8, -2) \]
Therefore, the midpoint of this line segment is \( (8, -2) \).

Coordinates

Coordinates are a pair of values that show an exact position on a plane. The coordinate system is typically defined using two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).

Each point in this system is represented as \((x, y)\), where \(x\) is the point's horizontal position and \(y\) is the vertical position. For example, in our exercise:

• The point \( P_1 \) has coordinates \( (7, -5) \), meaning it is 7 units to the right of the origin and 5 units down.
• The point \( P_2 \) has coordinates \( (9, 1) \), meaning it is 9 units to the right of the origin and 1 unit up.

When finding the midpoint of these two points, we are essentially finding the average position between them based on their coordinates. Thus, understanding coordinates and how they work is fundamentally important for locating points and solving many geometric problems.