Problem 134 - 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. $$\left(\frac{1}{3}, \frac{2}{3}\right),\left(-1,-\frac{3}{2}\right)$$

Step-by-Step Solution

Answer

The distance between the points \((\frac{1}{3}, \frac{2}{3})\) and \((-1,-\frac{3}{2})\) is calculated in step 2. The midpoint of the line segment connecting these points is found in step 3.

Step 1: Plot the Points

Plot the points \((\frac{1}{3}, \frac{2}{3})\) and \((-1,-\frac{3}{2})\) in a Cartesian coordinate system. The axes should be scaled with respect to the range of coordinates provided.

Step 2: Calculate the Distance Between the Points

The distance \(d\) between the points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Substitute \((x_1, y_1)\) with \((\frac{1}{3}, \frac{2}{3})\) and \((x_2, y_2)\) with \((-1,-\frac{3}{2})\) to get the distance.

Step 3: Find the Midpoint

The midpoint \((x_m, y_m)\) of a segment connecting the points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula: \(x_m = \frac{x_1 + x_2}{2}\), \(y_m = \frac{y_1 + y_2}{2}\). Substitute \((x_1, y_1)\) with \((\frac{1}{3}, \frac{2}{3})\) and \((x_2, y_2)\) with \((-1,-\frac{3}{2})\) to get the midpoint.

Key Concepts

Distance Formula

The distance formula is a vital tool in coordinate geometry for finding the straight-line distance between two points on a plane. Imagine you have two points,

• Point A with coordinates \((x_1, y_1)\)
• Point B with coordinates \((x_2, y_2)\)

You can calculate the distance \(d\) between them using the formula:
\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
To break it down further:

• Subtract the x-coordinates to find the horizontal distance.
• Subtract the y-coordinates to find the vertical distance.
• Square both results to eliminate negative values and apply the Pythagorean Theorem.
• Add the squared results and take the square root to determine the actual distance.

Using the points provided in the exercise, \((\frac{1}{3}, \frac{2}{3})\) and \((-1, -\frac{3}{2})\), you simply substitute these numbers into the formula to find the distance. This formula helps visualize the separation between two locations within a coordinate system. It’s much like measuring the length of the hypotenuse in a right triangle formed by these two points.

Midpoint Formula

Finding the midpoint between two points is crucial when you’re working with line segments in geometry. The midpoint is the point that lies exactly half-way between two given points. If you have two points:

• Point A \((x_1, y_1)\)
• Point B \((x_2, y_2)\)

You can find the midpoint \((x_m, y_m)\) using these simple formulas:
\[x_m = \frac{x_1 + x_2}{2}\]\[y_m = \frac{y_1 + y_2}{2}\]
This process involves:

• Adding the x-coordinates and dividing by 2, and
• Adding the y-coordinates and dividing by 2.

It's like averaging the coordinates from both points to find a middle point. From our example with points \((\frac{1}{3}, \frac{2}{3})\) and \((-1, -\frac{3}{2})\), you simply plug these coordinates into the formulas to determine the midpoint. Understanding the midpoint formula is useful for solving problems involving bisected segments or finding centers in various geometric contexts.

Plotting Points

Plotting points on a coordinate plane is one of the foundational skills in geometry. The coordinate plane consists of two perpendicular lines:

• An x-axis, which runs horizontally, and
• A y-axis, which runs vertically.

Each point on this plane can be described using a pair of numbers, called coordinates \((x, y)\), where x indicates the horizontal position and y indicates the vertical position. Here's how you plot a point: - Start at the origin, (0, 0), which is where the x-axis and y-axis intersect.- Move horizontally to the position specified by the x-coordinate.- Then, move vertically to the position specified by the y-coordinate.For the exercise, you plot the points \((\frac{1}{3}, \frac{2}{3})\) and \((-1, -\frac{3}{2})\) on this Cartesian grid. Plotting points accurately is crucial when calculating further parameters like midpoints and distances, as every calculation relies on these initial placements. It also helps visualize and solve more complex geometry problems.