Problem 16 - Contemporary Precalculus

Question

Find the distance between the two points and the midpoint of the segment joining them. $$(-2,3),(-3,2)$$

Step-by-Step Solution

Answer

Question: Determine the distance between the points $(-2,3)$ and $(-3,2)$, and find the midpoint of the segment joining them. Answer: The distance between the points $(-2,3)$ and $(-3,2)$ is $\sqrt{2}$, and the midpoint of the segment joining them is $(-\frac{5}{2}, \frac{5}{2})$.

Step 1: Identify the given points

We are given two points in the Cartesian plane: $$(-2,3)$$ and $$(-3,2)$$.

Step 2: Use the distance formula

The distance formula states that in order to find the distance between two points $$A(x_1, y_1)$$ and $$B(x_2, y_2)$$ in the Cartesian plane, we can use the formula: $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ Here, $$A(-2, 3)$$ and $$B(-3, 2)$$.

Step 3: Calculate the distance

Applying the distance formula for the given points $$A(-2, 3)$$ and $$B(-3, 2)$$, we get: $$D = \sqrt{(-3 - (-2))^2 + (2 - 3)^2}$$ $$D = \sqrt{(-1)^2 + (-1)^2}$$ $$D = \sqrt{1+1}$$ $$D=\sqrt{2}$$

Step 4: Use the midpoint formula

Now, let's find the midpoint between the same points. The midpoint formula states that for two points $$A(x_1, y_1)$$ and $$B(x_2, y_2)$$ in the Cartesian plane, the midpoint $$M(x, y)$$ can be found using the formula: $$x = \frac{x_1 + x_2}{2}$$ $$y = \frac{y_1 + y_2}{2}$$

Step 5: Calculate the midpoint

Applying the midpoint formula for the given points $$A(-2, 3)$$ and $$B(-3, 2)$$, we get: $$x = \frac{-2 + (-3)}{2} = \frac{-5}{2}$$ $$y = \frac{3 + 2}{2} = \frac{5}{2}$$ So, the midpoint $$M$$ is $$(-\frac{5}{2}, \frac{5}{2})$$. To summarize, the distance between the points $$(-2,3)$$ and $$(-3,2)$$ is $$\sqrt{2}$$, and the midpoint of the segment joining them is $$(-\frac{5}{2}, \frac{5}{2})$$.

Key Concepts

Midpoint Formula

The midpoint formula is a key tool used in analytic geometry to find the exact center point between two defined points on a Cartesian plane. This formula helps determine the point that lies exactly halfway along the line segment connecting these two points. It’s expressed as:

• \( x = \frac{x_1 + x_2}{2} \)
• \( y = \frac{y_1 + y_2}{2} \)

For example, if you have two points \((-2, 3)\) and \((-3, 2)\), you can apply the formula to find the midpoint \(M(x, y)\). Start by adding the respective coordinates and then divide each sum by 2:

• \( x = \frac{-2 + (-3)}{2} = \frac{-5}{2} \)
• \( y = \frac{3 + 2}{2} = \frac{5}{2} \)

Thus, the midpoint is \((-\frac{5}{2}, \frac{5}{2})\). This result represents the point that is equidistant from both original points.

Analytic Geometry

Analytic geometry, often called coordinate geometry, bridges algebra and geometry using a coordinate system. This connection allows us to solve geometrical problems in algebraic terms. One of the primary tools in analytic geometry includes finding distances and midpoints on the Cartesian plane.
It’s employed extensively to calculate various geometric elements like lengths of segments, equations of geometric shapes, and angles between lines. The methods of analytic geometry transform visual and spatial problems into algebraic ones, providing numerical answers to otherwise abstract questions.
While geometry typically involves the physical or graphical representation of shapes, analytic geometry allows these shapes to be analyzed through algebraic equations. This makes it possible to use equations to describe geometric figures, such as lines and circles, using the concepts of slopes and radii covered by algebra.

Cartesian Plane

The Cartesian plane is a two-dimensional plane defined by a pair of perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). It serves as a fundamental framework for many areas of mathematics, including analytic geometry. Each point on the Cartesian plane is identified by coordinates \((x, y)\), where \(x\) is the horizontal component and \(y\) is the vertical component.
This plane allows mathematicians to represent geometric shapes algebraically. By plotting points and drawing shapes on this plane, it becomes easier to visualize and solve problems related to distances, midpoints, and other geometric properties. For instance, by locating the points \((-2, 3)\) and \((-3, 2)\) on this plane, you can calculate their distance using the distance formula, or find their midpoint, enhancing the understanding of the spatial relationships between points.
The Cartesian plane is named after René Descartes, whose work laid the foundation for this system. It’s invaluable in both elementary and advanced mathematics as it provides a clear method to transition between geometric and algebraic perspectives.