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Chapter 2: Problem 34

Determine whether the lines through the pairs of points are parallel. $$ A(2,3), B(2,-2) \text { and } C(-2,4), D(-2,5) $$

Short Answer

Expert verified
The lines passing through points A(2,3) and B(2,-2) and through points C(-2,4) and D(-2,5) are both vertical lines with undefined slopes. Since vertical lines are always parallel, these two lines are parallel.

Step by step solution

01

Finding the slope of the line through points A and B

We are given points A(2,3) and B(2,-2). To find the slope of the line AB, we will use the slope formula: \(m = \frac{y2 - y1}{x2 - x1}\) Plugging in the coordinates of points A and B: \(m_{AB} = \frac{-2 - 3}{2 - 2}\)
02

Simplify the slope of the line through points A and B

We'll simplify the expression for the slope: \(m_{AB} = \frac{-5}{0}\) Since the denominator is zero in this case, the slope is undefined, meaning the line AB is a vertical line.
03

Finding the slope of the line through points C and D

Next, we need to find the slope of the line through points C(-2,4) and D(-2,5) using the same slope formula: \(m = \frac{y2 - y1}{x2 - x1}\) Plugging in the coordinates of points C and D: \(m_{CD} = \frac{5 - 4}{-2 - (-2)}\)
04

Simplify the slope of the line through points C and D

Now we'll simplify the expression for the slope: \(m_{CD} = \frac{1}{0}\) Again, since the denominator is zero, the slope is also undefined, meaning the line CD is a vertical line.
05

Compare the slopes

After calculating the slopes of both lines, we found that both are undefined, which means both lines are vertical. Vertical lines are always parallel, as they have the same slope. Therefore, the lines passing through points A and B and through points C and D are parallel.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope of a Line
The slope of a line is a measure that describes how steep the line is. It's calculated by the change in vertical direction (rise) divided by the change in horizontal direction (run). Mathematically, the slope is represented by:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]This formula helps us understand the direction and steepness of a line:
  • If the slope is positive, the line ascends from left to right.
  • If the slope is negative, the line descends from left to right.
  • If the slope is zero, the line is horizontal.
  • If the slope is undefined, the line is vertical.
In the exercise, calculating the slope through point pairs helps in determining the line's orientation. If you input the points' coordinates and notice that the denominator (\(x_2 - x_1\)) is zero, you encounter an undefined slope, leading us to a special type of line called a vertical line.
Undefined Slope
When dealing with lines in a coordinate plane, an undefined slope arises in certain situations. Specifically, when the difference in the x-values (run) between two points on the line is zero, the slope is undefined.
This occurs because dividing by zero doesn't work in mathematics. On a graph, this scenario presents as a vertical line.
A key takeaway about undefined slope:
  • It means the line is vertical.
  • It occurs when the x-coordinates of a line's endpoints are the same.
  • Vertical lines extend infinitely in the vertical direction.
When calculating the slope as in the given exercise, recognizing a zero denominator quickly identifies an undefined slope and saves time.
In essence, an undefined slope distinguishes vertical lines, which subsequently leads us to the concept of parallelism in vertical lines.
Vertical Lines
Vertical lines are unique in the world of geometry. Their defining feature is the characteristic vertical slope where all points on the line share the same x-coordinate, making these lines quite straightforward.
Vertical lines extend up and down without deviating side-to-side.
Some important characteristics include:
  • The slope is undefined because there is no horizontal change.
  • They are represented as \(x = a\), where \(a\) is the shared x-coordinate of all points.
  • Vertical lines are always parallel to each other.
In the exercise, recognizing both lines as vertical due to their undefined slope led to the conclusion that the lines are parallel. Thus, all vertical lines share this parallel trait, regardless of their position on the plane. This makes vertical lines predictable and easier to analyze in terms of their spatial relationships.

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Most popular questions from this chapter

The ratio of working-age population to the elderly in the United States (including projections after 2000 ) is given by $$ f(t)=\left\\{\begin{array}{ll} 4.1 & \text { if } 0 \leq t<5 \\ -0.03 t+4.25 & \text { if } 5 \leq t<15 \\ -0.075 t+4.925 & \text { if } 15 \leq t \leq 35 \end{array}\right. $$ with \(t=0\) corresponding to the beginning of 1995 . a. Sketch the graph of \(f\). b. What was the ratio at the beginning of 2005 ? What will be the ratio at the beginning of 2020 ? c. Over what years is the ratio constant? d. Over what years is the decline of the ratio greatest?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If \(a\) and \(c\) have opposite signs, then the parabola with equation \(y=a x^{2}+b x+c\) intersects the \(x\) -axis at two distinct points.

DECISION ANALYSIS A product may be made using machine I or machine II. The manufacturer estimates that the monthly fixed costs of using machine I are \(\$ 18,000\), whereas the monthly fixed costs of using machine II are \(\$ 15,000\). The variable costs of manufacturing 1 unit of the product using machine I and machine II are \(\$ 15\) and \(\$ 20\), respectively. The product sells for \(\$ 50\) each. a. Find the cost functions associated with using each machine. b. Sketch the graphs of the cost functions of part (a) and the revenue functions on the same set of axes. c. Which machine should management choose in order to maximize their profit if the projected sales are 450 units? 550 units? 650 units? d. What is the profit for each case in part (c)?

CRICKET CHIRPING AND TEMPERATURE Entomologists have discovered that a linear relationship exists between the rate of chirping of crickets of a certain species and the air temperature. When the temperature is \(70^{\circ} \mathrm{F}\), the crickets chirp at the rate of 120 chirps/min, and when the temperature is \(80^{\circ} \mathrm{F}\), they chirp at the rate of 160 chirps/min. a. Find an equation giving the relationship between the air temperature \(T\) and the number of chirps/min \(N\) of the crickets. b. Find \(N\) as a function of \(T\) and use this formula to determine the rate at which the crickets chirp when the temperature is \(102^{\circ} \mathrm{F}\).

MARKET EQUILIBRIUM The management of the Titan Tire Company has determined that the weekly demand and supply functions for their Super Titan tires are given by $$ \begin{array}{l} p=144-x^{2} \\ p=48+\frac{1}{2} x^{2} \end{array} $$respectively, where \(p\) is measured in dollars and \(x\) is measured in units of a thousand. Find the equilibrium quantity and price.
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