Problem 96 - Algebra and Trigonometry

Question

Find the midpoint of the line segment connecting the points (-3,7) and \(\left(\frac{1}{2}, 2\right)\).

Step-by-Step Solution

Answer

Midpoint = \left( \frac{-5}{4}, \frac{9}{2} \right).

Step 1: Understand the Midpoint Formula

The midpoint of a line segment connecting two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) is given by the formula: \(\text{Midpoint} = \frac{(x_1 + x_2)}{2}, \frac{(y_1 + y_2)}{2}\).

Step 2: Identify Coordinates of Given Points

Identify the coordinates of the given points: \(A(-3, 7)\) and \(\frac{1}{2}, 2\). Here, \(x_1 = -3\), \(y_1 = 7\), \(x_2 = \frac{1}{2}\), and \(y_2 = 2\).

Step 3: Plug Coordinates into the Midpoint Formula

Substitute \((x_1, y_1) = (-3, 7)\) and \((x_2, y_2) = \frac{1}{2}, 2\) into the midpoint formula: \((\text{Midpoint}) = \frac{((-3) + (\frac{1}{2}))}{2}, \frac{(7 + 2)}{2}\).

Step 4: Calculate the Midpoint

Perform the calculations: \(\frac{((-3) + (\frac{1}{2}))}{2} = \frac{(-3 + \frac{1}{2})}{2} = \frac{\frac{-6 + 1}{2}}{2} = \frac{\frac{-5}{2}}{2} = \frac{-5}{4}\). Similarly, \(\frac{(7+2)}{2} = \frac{9}{2}\). So, the midpoint is \left( \frac{-5}{4}, \frac{9}{2} \right).

Key Concepts

coordinate geometry

Coordinate geometry, also known as analytic geometry, bridges algebra and geometry. It allows us to study geometric figures using a coordinate system. By plotting points, lines, and shapes on a plane defined by an x-axis and a y-axis, we can solve various geometric problems using algebraic methods. This approach is essential for understanding the spatial relationships and properties of geometric figures. By using coordinates to define points, we can easily calculate distances, slopes, and midpoints, making abstract geometric concepts more tangible and understandable.

midpoint calculation

Midpoint calculation is a fundamental concept in coordinate geometry. The midpoint of a line segment is the point that divides the segment into two equal halves. To find the midpoint, we use the midpoint formula: \(\text{Midpoint} = \frac{(x_1 + x_2)}{2}, \frac{(y_1 + y_2)}{2}\). This formula takes the average of the x-coordinates and the y-coordinates of the two endpoints of the line segment.

For the points A(-3, 7) and \(\frac{1}{2}, 2\), the midpoint is calculated by substituting the coordinates into the formula: \(\text{Midpoint} = \frac{(-3 + \frac{1}{2})}{2}, \frac{(7 + 2)}{2}\). Performing the calculations step-by-step, we get:

• \(\frac{(-3 + \frac{1}{2})}{2} = \frac{-5}{4}\)
• \(\frac{(7 + 2)}{2} = \frac{9}{2}\)

Therefore, the midpoint is \(\frac{-5}{4}, \frac{9}{2}\).

line segment

A line segment is a part of a line that is bounded by two distinct endpoints. Unlike a line, which extends infinitely in both directions, a line segment has a definite start and end. Line segments are fundamental in geometry and are used to construct various shapes and figures.

Considering two points A and B, the line segment connecting A(-3, 7) and B(\frac{1}{2}, 2) is an example. The length of this line segment can be calculated using the distance formula, but for midpoint calculations, we focus on finding the exact middle point which divides the segment into two equal halves.
By applying the midpoint formula, we have found that the midpoint of the segment connecting A and B is \(\frac{-5}{4}, \frac{9}{2}\). This midpoint splits the line segment into two equal parts, making it simpler to analyze the geometrical properties of the segment.