Problem 47 - 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. \( (1, 1) \), \( (9, 7) \)

Step-by-Step Solution

Answer

The plotted points are \( (1, 1) \) and \( (9, 7) \). The distance between these points is 10 units. The midpoint of the line segment joining these points is \( (5, 4) \).

Step 1: Plotting the Points

The first step to solve this problem is to plot the two given points, \( (1, 1) \) and \( (9, 7) \), on the coordinate plane.

Step 2: Find the distance

Finding the distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) in a plane is governed by the formula \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). In this case, \( x_1 = 1\), \( y_1 = 1\), \( x_2 = 9\), and \( y_2 = 7\). So, the distance between the points can be obtained as \(\sqrt{(9-1)^2+(7-1)^2}\). Simplifying, we get the distance as \(\sqrt{8^2+6^2} = \sqrt{100} = 10\).

Step 3: Find the midpoint

The midpoint between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) can be found using the formula \(( (x_1 + x_2)/2 , (y_1 + y_2)/2 )\). Here, \( x_1 = 1\), \( y_1 = 1\), \( x_2 = 9\), and \( y_2 = 7\). Therefore, the midpoint of the line segment joining the points is \(( (1 + 9)/2 , (1 + 7)/2 ) = (5, 4)\).

Key Concepts

Midpoint Formula

The midpoint formula is a key concept when working with line segments in a coordinate plane. It helps us find the exact center point between two endpoints. The formula is represented as follows: \[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]This involves simply averaging the x-coordinates and y-coordinates of the given points. For instance, given points \((x_1, y_1)\) and \((x_2, y_2)\):

• Add the x-coordinates: \(x_1 + x_2\)
• Add the y-coordinates: \(y_1 + y_2\)
• Divide the sum of each by 2 to find the midpoint coordinates.

In our example, the points \((1, 1)\) and \((9, 7)\) yield a midpoint of \((5, 4)\). This point is precisely halfway between the two points on the coordinate plane.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can locate points defined by a pair of numerical coordinates. It's essential for visualizing mathematical problems, such as finding distances and midpoints. This plane consists of two perpendicular axes:

• The horizontal axis is labeled as the \(x\)-axis.
• The vertical axis is labeled as the \(y\)-axis.

The point where these axes intersect is known as the origin, denoted by \((0, 0)\). Each point plotted on the plane is represented by an ordered pair \((x, y)\), where \(x\) is the horizontal coordinate and \(y\) is the vertical coordinate. By understanding how to navigate this plane, one can easily apply formulas like the distance and midpoint to solve geometrical problems.

Plotting Points

Plotting points is a fundamental skill required to visually interpret problems on a coordinate plane. Each point represents a location defined by an ordered pair \((x, y)\). Let's break down how you can plot points effectively:

• Begin at the origin \((0, 0)\).
• Move horizontally to the \(x\)-coordinate. For instance, for point \((1, 1)\), move one unit right.
• From your \(x\) position, move vertically to the \(y\)-coordinate. Continuing our example, move up one unit.

Repeat this process for multiple points. Accurate plotting is crucial as it sets the stage for applying other concepts like distance and midpoint. With practice, plotting will become intuitive, allowing deep insights into mathematical geometry.