Question

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

Short answer

The midpoint is \left( \frac{a}{2}, \frac{a}{2} \right).

Step-by-step solution

Identify the Coordinates

First note the coordinates of the points. Point \(P_{1}\) has coordinates \( (a, a) \) and point \(P_{2}\) has coordinates \( (0, 0) \).

Use the Midpoint Formula

The formula for the midpoint \( M(x_m, y_m) \) between two points \(P_{1}(x_{1}, y_{1}) \) and \( P_{2}(x_{2}, y_{2}) \) is given by: \ \[ M = \left( \frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2} \right) \]

Substitute the Coordinates

Substitute the coordinates of \(P_{1} \) and \(P_{2} \) into the midpoint formula: \ \[ M = \left( \frac{a + 0}{2}, \frac{a + 0}{2} \right) = \left( \frac{a}{2}, \frac{a}{2} \right) \]

Key concepts

Midpoint Formula

The midpoint formula is a key concept in coordinate geometry used to find the exact center point between two endpoints of a line segment.

When you have two points, let’s call them \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\), the midpoint \((M)\) is calculated using the following formula:
\[ M = \left( \frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2} \right) \] This formula takes the average of the x-coordinates and the y-coordinates of the two points.
Its simplicity makes it a handy tool for solving various problems in geometry, such as finding the center of a line segment on a coordinate plane.

Let's consider an example: If you have points \((2, 3)\) and \((4, 7)\), the midpoint can be found by:
\[ M = \left( \frac{2 + 4}{2}, \frac{3 + 7}{2} \right) = \left( 3, 5 \right) \] So, the midpoint of the line segment joining these points is \((3, 5)\).

Coordinate Geometry

Coordinate geometry, often called analytic geometry, integrates algebra and geometry using a coordinate system.

Its fundamental notion is plotting points on an X-Y coordinate plane.
Through these points, one can form various shapes, lines, and curves, making it easier to prove different geometrical theorems algebraically.

In coordinate geometry, points are defined using ordered pairs \((x, y)\), representing their positions on the plane.
Lines can be defined using equations, slopes, and intercepts, while distance and midpoint formulas are used to analyze and compute distances and central points between points.

Practicing coordinate geometry helps develop spatial reasoning and problem-solving skills, making it valuable in fields like engineering, physics, and computer graphics.

Line Segment Midpoint

A line segment is a part of a line with two endpoints, and a midpoint is the exact middle point of this segment.

In the original exercise, the line segment joins the points \((a, a)\) and \((0, 0)\).

To find the midpoint, you’ll use the given midpoint formula:
\[ M = \left( \frac{a + 0}{2}, \frac{a + 0}{2} \right) = \left( \frac{a}{2}, \frac{a}{2} \right) \] This means the midpoint lies at \((\frac{a}{2}, \frac{a}{2})\).

This concept helps evenly divide a line segment, aiding in dividing tasks like finding centers in triangles or other polygons, and is foundational for many geometric constructions and proofs.