Question

\(A(-3,2)\) and \(B(5,4)\) are the end points of a line segment, find the coordinates of the mid points of the line segment : (a) \((1,3)\) (b) \((2,3)\) (c) \((3,2)\) (d) \((4,3)\)

Short answer

Answer: (1, 3)

Step-by-step solution

Identify the coordinates of the endpoints

The given endpoints of the line segment are \(A(-3,2)\) and \(B(5,4)\). So, \((x_1, y_1) = (-3,2)\) and \((x_2, y_2) = (5,4)\).

Apply the midpoint formula

Now, we will apply the midpoint formula to find the coordinates of the midpoint: Midpoint = \(\left(\frac{-3 + 5}{2}, \frac{2 + 4}{2}\right)\)

Solve for the coordinates of the midpoint

Calculate the midpoint coordinates: Midpoint = \(\left(\frac{2}{2}, \frac{6}{2}\right) = (1, 3)\)

Choose the correct option

Since the calculated midpoint is \((1, 3)\), the correct answer is option (a). So, the coordinates of the midpoint of the line segment with endpoints \(A(-3,2)\) and \(B(5,4)\) is \((1,3)\).

Key concepts

Coordinate Geometry

Coordinate geometry is a branch of mathematics that uses algebraic equations to describe geometric principles. It allows us to explore shapes and their properties through a coordinate plane. Imagine a grid, like graph paper. Each point on this grid is represented by a pair of numbers, known as coordinates, which describe its position. These coordinates tell us how far the point is from the horizontal (x-axis) and vertical (y-axis) lines.

In the context of a line segment, the endpoints are key. Each endpoint has its own set of coordinates that define its position. For example, in the exercise, the endpoints are given as

• \(A(-3, 2)\)
• \(B(5, 4)\)

With these coordinates, we can find various properties of the line segment, such as its length and midpoint. Understanding this connection between algebra and geometry enables us to solve practical problems with math.

Line Segment

A line segment is a part of a line with two endpoints. Unlike a line, which extends infinitely, a line segment has a definite beginning and end. This makes it possible to measure its length and identify its midpoint. Visualize a connecting line between two dots on a page; that’s your line segment.

In coordinate geometry, a line segment is essential because it forms the basis for many geometric constructs. By defining the endpoints, such as

• \(A(-3, 2)\)
• \(B(5, 4)\)

we can apply mathematical formulas to find further details like its midpoint or slope. In daily life, concepts of line segments help in fields such as engineering and computer graphics.

Mathematics Problem Solving

Mathematics problem-solving involves a systematic approach to finding solutions using logic and known formulas. The exercise focuses on using the midpoint formula to determine the center point of a line segment.

The midpoint formula is: \[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]This formula lets us find the exact middle point of the line segment by averaging the coordinates of the endpoints. Applying it to our endpoints:

• \((x_1, y_1) = (-3, 2)\)
• \((x_2, y_2) = (5, 4)\)

we calculate:

• \(\left( \frac{-3 + 5}{2}, \frac{2 + 4}{2} \right) = (1, 3)\)

The process not only involves calculating but also verifying that each step is logical. This step-by-step strategy is crucial for problem-solving in mathematics, ensuring you understand every part of the solution.