Problem 51 - Precalculus with Limits

Question

In Exercises 47-56, (a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. \( (-1, 2) \), \( (5, 4) \)

Step-by-Step Solution

Answer

The distance between the points is approximately 6.32 units and the midpoint of the line segment joining the points is \(\(2, 3\)\).

Step 1: Plotting the Points

Plot the points \(-1, 2\) and \(5, 4\) on a Cartesian plane. Note the position of these points, as they'll help in understanding the context of the problem and steps that follow.

Step 2: Calculating the Distance Between the Points

Use the distance formula which is derived from Pythagoras' theorem, \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Here, \(-1\) and \(2\) are \(x_1\) and \(y_1\) and \(5\) and \(4\) are \(x_2\) and \(y_2\) respectively. Substituting these values will provide the distance between these two points.

Step 3: Calculating the Midpoint of the Line Segment

Use the midpoint formula, \((\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})\). Here, \(-1\) and \(2\) are \(x_1\) and \(y_1\) and \(5\) and \(4\) are \(x_2\) and \(y_2\) respectively. Substituting these values will yield the location of the midpoint of the line segment joining the two points.

Key Concepts

Midpoint Formula

The midpoint formula is a useful tool in geometry. It's used to find the point exactly halfway between two other points on a Cartesian plane.
Imagine a line segment that connects the two points \((-1, 2)\) and \((5, 4)\). To find the midpoint of this line segment, you apply the midpoint formula: \[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \] Here, \(x_1\) and \(y_1\) are the coordinates of the first point, and \(x_2\) and \(y_2\) are the coordinates of the second point.
So, substitute the given values into the formula: \[ M = \left(\frac{-1 + 5}{2}, \frac{2 + 4}{2}\right) = \left(\frac{4}{2}, \frac{6}{2}\right) = (2, 3) \] This results in the midpoint \((2, 3)\).
What this means is that if you travel from \((-1, 2)\) to \((5, 4)\), \((2, 3)\) is the exact halfway point on your journey.

Plotting Points

Plotting points on a Cartesian plane is a foundational skill in mathematics. It helps visualize relationships between numbers and understand spatial concepts better.
When you plot a point like \((-1, 2)\), it involves:

• Moving left or right according to the x-value. For \(-1\), you move left, since it's negative.
• Moving up or down according to the y-value. For \(+2\), it's up.

Similarly, to plot \((5, 4)\), you would:

• Move 5 units to the right (since it's positive).
• Move 4 units up (again, positive value).

By marking these positions, you're effectively drawing an invisible line on the graph-paper-like grid of the Cartesian plane. It's crucial to get accurate plots because they set up every further calculation, such as those needed for distance or midpoints.

Cartesian Plane

The Cartesian plane is a powerful mathematical tool used for plotting and analyzing points in space. It's essentially a flat surface made up of two lines that intersect at right angles.
The horizontal line is called the x-axis, and it runs left to right. The vertical line is the y-axis, going up and down.
The point where these axes intersect is called the origin, labeled as \(0, 0\).
Each point on the plane is defined by a pair of numbers written as \(x, y\).
This unique system allows each point to be placed in a specific location:

• The x-value shows how far from the y-axis the point is horizontally.
• The y-value indicates the point's vertical distance from the x-axis.

Using the Cartesian plane, complex mathematical and real-world problems can be visualized and solved. For example, by plotting points like \((-1, 2)\) and \((5, 4)\), you can easily calculate distances or find midpoints.