Problem 29 - Algebra 1: Concepts and Skills

Question

Find the midpoint of the line segment connecting the given points. Then show that the midpoint is the same distance from each point. \((-3,-2),(1,7)\)

Step-by-Step Solution

Answer

The midpoint of the line segment connecting the points \(-3,-2\) and \(1,7\) is \(-1,2.5\). And, it's verified that this point is equidistant from both original points with a distance of approximately 4.93 units.

Step 1: Calculate the Midpoint

Use the midpoint formula \((x_1 + x_2)/2 , (y_1 + y_2)/2\) on the coordinates \(-3,-2\) and \(1,7\). Plug the values into the formula to get \((-3 + 1)/2 , (-2 + 7)/2\) which simplifies to \(-2/2 , 5/2\), which results in the point \(-1, 2.5\).

Step 2: Calculate Distances to the Midpoint

Use the distance formula \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) to find the distances from the midpoint to the two original points. Using the points \(-3,-2\) and \(-1,2.5\) in the formula gives \(\sqrt{(-1 - (-3))^2 + (2.5 - (-2))^2} = \sqrt{4 + 20.25} = \sqrt{24.25} = 4.93\). Similarly, for the points \(1,7\) and \(-1,2.5\) the distance is \(\sqrt{(-1 - 1)^2 + (2.5 - 7)^2} = \sqrt{4 + 20.25} = \sqrt{24.25} = 4.93\).

Step 3: Verify the Distances

Compare the calculated distances. If they're equal, the point is indeed a midpoint. Here, both distances are 4.93, proving that \(-1, 2.5\) is the midpoint.

Key Concepts

Distance Formula

The distance formula is a powerful tool in geometry that helps you find the length between two points in a coordinate plane. It is derived from the Pythagorean Theorem, which you might know from right triangle studies. To apply the distance formula, consider two points

• Point 1: \(x_1, y_1\)
• Point 2: \(x_2, y_2\)

You can determine the distance between these two points using the equation: \[\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
You simply subtract the x-coordinates and y-coordinates separately, square the results, and sum them. The square root of that sum gives you the distance.
In practical terms, this formula is essential in problems where you need to verify or compare distances, such as confirming that a point is equidistant from two other points, like identifying midpoints.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry through a coordinate system. This system allows us to represent geometric figures like points, lines, and shapes in a two-dimensional space using ordered pairs \(x, y\). For instance, to define a line segment, you need the coordinates of its endpoints.
Coordinate geometry enables you to explore geometric properties and relationships algebraically.

• This makes it easier to compute lengths using the distance formula.
• It also allows the calculation of other properties, like slope and midpoint, with algebraic formulas.

The framework of coordinate geometry provides a comprehensive method to solve geometric problems analytically. It turns what might otherwise be a difficult geometric task into a manageable algebraic calculation, while allowing visual interpretation and testing the accuracy of solutions.

Line Segment

A line segment is a part of a line bounded by two distinct endpoints. Unlike a line, it does not extend infinitely in either direction.
When dealing with line segments in coordinate geometry, both endpoints are defined by their coordinates, which helps in making precise calculations.
The concept of a midpoint plays an integral role in studying line segments. For any two points defining a line segment, \(A(x_1, y_1)\text{ and } B(x_2, y_2)\), the midpoint is calculated using the formula:\[\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\]This point is equidistant from both endpoints, which we verify using the distance formula. Understanding line segments involves recognizing these relationships and using them to solve problems like subdividing a segment or proving congruence.