Problem 133 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment connecting the points. $$\left(-1, \frac{1}{2}\right),\left(\frac{4}{3}, \frac{5}{2}\right)$$

Step-by-Step Solution

Answer

The points \(-1, \frac{1}{2}\) and \(\frac{4}{3}, \frac{5}{2}\) are plotted on the Cartesian plane. The calculated distance between these points using the distance formula is \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\), and the midpoint coordinates are obtained using the midpoint formula \((\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})\).

Step 1: Plotting the points

Plot the two given points \(-1, \frac{1}{2}\) and \(\frac{4}{3}, \frac{5}{2}\) on the Cartesian plane. The first value of the pair gives the x-coordinate and the second gives the y-coordinate.

Step 2: Calculate the distance

Use the distance formula \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\). Substitute \(x_1 = -1\), \(y_1 = \frac{1}{2}\), \(x_2 = \frac{4}{3}\), and \(y_2 = \frac{5}{2}\). Calculate to find the distance.

Step 3: Find the midpoint

Use the midpoint formula \((\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})\). Substitute \(x_1 = -1\), \(y_1 = \frac{1}{2}\), \(x_2 = \frac{4}{3}\), and \(y_2 = \frac{5}{2}\). Computing this gives the coordinates of the midpoint.

Key Concepts

Distance Formula

The distance formula in coordinate geometry is a tool to calculate how far apart two points are on a Cartesian plane. Imagine these points as ends of a straight line, and the distance formula helps us find the length of that line. It is derived from the Pythagorean theorem.
Here’s the formula:

• \[\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

To use this formula, you need two points. Let's label them as

• \((x_1, y_1)\)
• \((x_2, y_2)\)

. The distance is then calculated by subtracting the x-coordinates and the y-coordinates of the two points, squaring these results, adding them, and taking the square root of the sum. It's as simple as plug and play.
For example, let’s work with the points \((-1, \frac{1}{2})\) and \(\left(\frac{4}{3}, \frac{5}{2}\right)\). Plug these into the formula:
\[\sqrt{\left(\frac{4}{3} + 1\right)^2 + \left(\frac{5}{2} - \frac{1}{2}\right)^2}\]Simplifying this, we find the actual distance between these points on the plane. Understanding this formula is crucial for geometric and algebraic calculations, especially when dealing with real-world measurements.

Midpoint Formula

The midpoint formula is used to find the point that is exactly halfway between two given points on a Cartesian plane. Think of it as finding the balance point on a seesaw. It provides a neat and predictable way to find this middle location.
Here's how it works:

• \[\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\]

The formula uses averages of the x-coordinates and the y-coordinates of two points. Simply add the x-coordinates and divide by two to get the x-value of the midpoint. Do the same for the y-coordinates to get the y-value.
Using our example points \((-1, \frac{1}{2})\) and \(\left(\frac{4}{3}, \frac{5}{2}\right)\), we can plug these numbers into the formula:
\[\left(\frac{-1 + \frac{4}{3}}{2}, \frac{\frac{1}{2} + \frac{5}{2}}{2}\right)\]This helps us find the midpoint of the line segment between these two points. This concept is widely used in navigation, architecture, and even sports analytics, wherever you need to evenly divide a space.

Cartesian Plane

The Cartesian plane is a two-dimensional surface that allows us to graphically interpret algebraic equations and visualize geometric figures. It is named after the mathematician René Descartes, who developed the concept. This plane has two number lines:

• Horizontal (x-axis)
• Vertical (y-axis)

They intersect at a point called the origin. The Cartesian plane divides space into four quadrants, each of which can hold positive or negative values of x and y.
When plotting a point on this plane, the x-coordinate tells you how far right or left the point is from the origin, and the y-coordinate tells you how far up or down the point is. For example, the point \((-1, \frac{1}{2})\) is placed 1 unit left from the origin and half a unit up.
The Cartesian plane serves as the chalkboard for many mathematical and scientific applications, aiding in plotting data, designing structures, and simulating movements.