Problem 25 - College Algebra

Question

Find the midpoint of each line segment with the given endpoints. $$\left(-\frac{7}{2}, \frac{3}{2}\right)\( and \)\left(-\frac{5}{2},-\frac{11}{2}\right)$$

Step-by-Step Solution

Answer

The midpoint of the line segment with the given endpoints is (-3, -2).

Step 1: Identify the coordinates of the endpoints

The coordinates of the first end point A are \(-\frac{7}{2}\), \frac{3}{2}\ and of the second end point B are \(-\frac{5}{2}\), \(-\frac{11}{2}\).

Step 2: Apply the midpoint formula

The midpoint M can be found by using the formula M= \(\left(\frac{x1 + x2}{2}, \frac{y1 + y2}{2}\right)\). So, M=\(\left(\frac{(-\frac{7}{2}) + (-\frac{5}{2})}{2}, \frac{\frac{3}{2}+ (-\frac{11}{2})}{2}\right)\).

Step 3: Calculate the coordinates of the midpoint

We perform the arithmetic in the formula to obtain: M=\(\left(\frac{-7 - 5}{4}, \frac{3 - 11}{4}\right)\) = \(\left(\frac{-12}{4}, \frac{-8}{4}\right)\) = (-3, -2).

Key Concepts

Coordinates of Endpoints

Understanding the coordinates of endpoints is crucial when dealing with problems in geometry and algebra, especially involving line segments. To identify the endpoints of a line segment, one must know the x (horizontal) and y (vertical) values that represent a point in a two-dimensional plane. An endpoint's coordinates are typically written in the form (x, y).

For instance, in our exercise example, the endpoints of the line segment are given as \(\left(-\frac{7}{2}, \frac{3}{2}\right)\) and \(\left(-\frac{5}{2},-\frac{11}{2}\right)\). These pairs of numbers tell us the exact location of the endpoints on a coordinate plane, essential for further calculations like finding the segment’s midpoint. When performing calculations, it's important to pay attention to the signs (positive or negative) of each coordinate, as they determine the direction from the origin (0, 0) on the grid.

College Algebra

College algebra serves as the foundation for a variety of mathematical concepts and is often where students first encounter problems like finding the midpoint of a line segment. It typically includes a study of functions, equations, inequalities, and the use of algebraic methods to solve real-world problems.

In the context of our exercise, college algebra provides the tools you need to manipulate and simplify algebraic expressions. By understanding algebraic operations and properties of numbers, students can tackle the midpoint formula effectively. Being proficient in algebra is not only about solving equations; it also helps in understanding how mathematical relationships work, which is vital for higher-level math and many scientific fields.

Arithmetic in Algebra

The term ‘arithmetic in algebra’ refers to the use of basic arithmetic operations—addition, subtraction, multiplication, and division—within algebraic expressions and equations. These operations are the building blocks for more complex algebraic concepts.

For example, to find the midpoint of the line segment in our exercise, you literally apply arithmetic within the algebraic midpoint formula. The midpoint M is the average of the x-coordinates and the y-coordinates of the endpoints, calculated as \(M=\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\). This requires adding the corresponding coordinates of the endpoints together and then dividing by 2—an application of both addition and division. Mastery of arithmetic is thus essential for success in algebra because it allows for the manipulation of these expressions into a more simplified form, leading to a solution like \(M=(-3, -2)\) in our given problem.