Problem 25 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The midpoint of the line segment with endpoints \( A\left(-\frac{7}{2}, \frac{3}{2}\right)\) and \( B\left(-\frac{5}{2},-\frac{11}{2}\right) \) is \( M(-3, -2) \).

Step 1: Identify the Coordinates

The coordinates of the given points are \( A\left(-\frac{7}{2}, \frac{3}{2}\right)\) and \( B\left(-\frac{5}{2},-\frac{11}{2}\right) \). Therefore, \( x_1 = -\frac{7}{2}, y_1 = \frac{3}{2}, x_2 = -\frac{5}{2} \), and \( y_2 = -\frac{11}{2} \).

Step 2: Apply the Midpoint Formula

Substitute the coordinates into the midpoint formula to get: \( M = \left(\frac{-\frac{7}{2} -\frac{5}{2}}{2}, \frac{\frac{3}{2} -\frac{11}{2}}{2}\right)\).

Step 3: Solve the Formulas

Simplify the resulting equations to find the x and y coordinates of the midpoint: \( M = \left(\frac{-\frac{12}{2}}{2}, \frac{-\frac{8}{2}}{2}\right) = \left(-3, -2\right)\).

Key Concepts

Coordinates of a Point

To understand the geometry of shapes and the layout of graphs, it's crucial to comprehend the concept of coordinates of a point. Coordinates are a set of values that show an exact position on a two-dimensional plane, typically marked on an x (horizontal) and y (vertical) axis. For instance, the coordinates \( -\frac{7}{2}, \frac{3}{2} \) correspond to a point that is \( -\frac{7}{2} \) units along the x-axis (to the left of the origin if negative) and \( \frac{3}{2} \) units along the y-axis (above the origin if positive).

When plotting points, always start at the center, known as the origin, and move horizontally first, followed by vertically. In the exercise provided, the points have negative and positive coordinates, suggesting one point lies in the second quadrant (since both coordinates are negative) and the other point straddles the first and second quadrants (since x is negative and y is positive).

• The first coordinate, x, determines left or right movement from the origin.
• The second coordinate, y, dictates the up or down traversal.
• To plot negative values, move left (for x) or down (for y) from the origin.
• To plot positive values, move right (for x) or up (for y).

Midpoint of a Line Segment

The midpoint of a line segment is the point that divides the segment into two congruent parts. It’s like the ‘halfway’ point. Finding the midpoint is helpful in various branches of mathematics, including geometry, algebra, and precalculus. The standard method for determining a midpoint's coordinates is using the midpoint formula: \( M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \), where \( x_1 \) and \( y_1 \) are the coordinates of the first point and \( x_2 \) and \( y_2 \) the coordinates of the second point.

In our exercise, we applied this formula to the points \( A(-\frac{7}{2}, \frac{3}{2}) \) and \( B(-\frac{5}{2}, -\frac{11}{2}) \).

Average of X and Y Coordinates

By simply averaging the x-coordinates and the y-coordinates separately, we find where they meet to form the midpoint. This is a crucial concept as it represents a geometrically important center point between the two, which could be used in further analytical geometry problems or in practical applications such as computer graphics, navigation, and construction.

Precalculus

When dealing with mathematics at the precalculus level, you're in the realm that prepares you for the complexity of calculus itself. Precalculus involves a variety of topics, such as functions, sequences, series, trigonometry, and analytic geometry – which includes our study of coordinates and finding midpoints. Developing a solid understanding of precalculus is foundational for tackling the more advanced challenges in calculus.

• Precalculus deepens understanding of algebraic and geometric concepts.
• It bridges the gap between algebra/geometry and calculus.
• Problems like finding a line segment's midpoint are typical examples of precalculus exercises.

Grasping these precalculus concepts ensures that students are primed for the logical reasoning and problem-solving skills crucial for success in calculus. In our line segment midpoint exercise, we're applying precalculus knowledge to solve real-world problems by calculating a precise location geometrically shared between two points.