Problem 39 - Introductory and Intermediate Algebra for College Students 4th

Question

Use slopes to solve Exercises \(39-40\). Show that the points whose coordinates are \((-3,-3)\) \((2,-5),(5,-1),\) and \((0,1)\) are the vertices of a four-sided figure whose opposite sides are parallel. (Such a figure is called a parallelogram.)

Step-by-Step Solution

Answer

If the slopes of opposite sides of the four-sided figure are the same, this verifies that the opposite sides are parallel and the given figure is indeed a parallelogram.

Step 1: Calculate the Slopes

First, we need to find the slope of the lines connecting each pair of vertices. The slope of a line connecting two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the slope formula \(m = \frac{(y_2 - y_1) } { (x_2 - x_1) }\). So, using this formula, calculate the slopes for lines connecting points \((-3,-3)\) and \((2,-5)\), \((0,1)\) and \((5,-1)\), \((-3,-3)\) and \((0,1)\), and \((2,-5)\) and \((5,-1)\).

Step 2: Compare the Slopes

Next, compare the slopes that you calculated in the first step. If the slopes of opposite sides are equal, then the opposite sides are parallel. This means that the four-sided figure is a parallelogram.

Step 3: Conclusions

After comparing the slopes, if the opposite sides have equal slopes, the conclusion can be drawn that the figure whose vertices are given is a parallelogram.

Key Concepts

Slope Formula

The slope formula is your go-to tool when you want to determine the steepness or inclination of a line connecting two points on a plane.It's the mathematical magic that helps us find out how two points relate linearly. Given two points \( (x_1, y_1) \) and \( (x_2, y_2) \), the slope \( m \) is calculated using the formula:

• \( m = \frac{(y_2 - y_1) }{ (x_2 - x_1) } \)

This formula tells us how much the y-value changes for each unit change in the x-value.Such a number indicates whether the line is rising, falling, or perhaps staying constant as you move along it. For instance, if the slope is positive, the line rises as it moves from left to right. If the slope is negative, the line falls. A zero slope means the line is perfectly horizontal, while an undefined slope suggests a vertical line.
Knowing how to calculate the slope is crucial, especially in solving exercises involving parallelograms, as it helps identify parallel lines by comparing slopes from different line segments on the figure.

Parallel Lines

Parallel lines are lines in a plane that, no matter how far extended, do not meet. Imagine two railway tracks running alongside each other. That's a simple visualization of parallel lines.In terms of slopes, two lines are parallel if and only if they have the same slope. So, if you calculate the slope of one line connecting two points and find it to be \( m \), any line parallel to it will also have a slope of \( m \).
For our given exercise, this is precisely what we need to identify a parallelogram. A parallelogram is a quadrilateral (four-sided figure) where both pairs of opposite sides are parallel. Thus, by calculating the slopes of the opposite sides and proving they are equal, it confirms the defining property of a parallelogram. Recognizing these characteristic traits through slope comparison is a powerful application of understanding parallel lines.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, bridges algebra and geometry using a coordinate system. It mainly involves plotting points, lines, and shapes on a graph and using algebra to understand the properties of these geometric figures.
In coordinate geometry, you can determine various characteristics of shapes, such as length, midpoint, and slope, which we explored in our problem.

• Coordinates give a precise location of points.
• The slope formula helps determine inclinations and directions of lines.

In the context of this problem, by plotting the vertices \((-3,-3), (2,-5), (5,-1), \text{and}\ (0,1)\),you can visualize the figure and use calculations to establish relationships between sides (such as parallelism). These applications of coordinate geometry clarify complex figures like parallelograms, enabling us to draw conclusions about their properties easily. It's a vital tool in geometry, as it simplifies concepts by converting them into algebraic equations, making it easier to solve geometric problems.