Problem 44 - Contemporary Precalculus

Question

Determine whether the line through \(P\) and \(Q\) is parallel or perpendicular to the line through \(P=(0,3 / 2), Q=(1,1)\) and \(R=(2,7), S=(3,9)\)

Step-by-Step Solution

Answer

Question: Determine whether the line passing through points P(0, 3/2) and Q(1, 1) is parallel or perpendicular to the line passing through points R(2, 7) and S(3, 9). Answer: The line passing through points P and Q is perpendicular to the line passing through points R and S.

Step 1: Calculate the slope of line PQ

To find the slope of the line passing through points P and Q, we use the formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\) Let P be \((x_1, y_1) = (0, \frac{3}{2})\) and Q be \((x_2, y_2) = (1, 1)\). Then the slope of the line PQ is: \(m_{PQ} = \frac{1 - \frac{3}{2}}{1 - 0} = -\frac{1}{2}\)

Step 2: Calculate the slope of line RS

Using the same formula as in step 1, let R be \((x_1, y_1) = (2, 7)\) and S be \((x_2, y_2) = (3, 9)\). Then the slope of line RS is: \(m_{RS} = \frac{9 - 7}{3 - 2} = 2\)

Step 3: Determine if the lines are parallel or perpendicular

Now that we have the slopes of both lines, we will check if they are parallel or perpendicular. Lines are parallel if their slopes are equal (\(m_{PQ} = m_{RS}\)), which is not the case in our problem. Lines are perpendicular if the product of their slopes is equal to -1 (\(m_{PQ} \times m_{RS} = -1\)). In our case, the product of \(-\frac{1}{2}\) and \(2\) is \((-\frac{1}{2}) \times 2 = -1\), so the lines PQ and RS are indeed perpendicular.

Key Concepts

Slope of a Line

The slope of a line is a measure of its steepness and direction. It calculates the vertical change divided by the horizontal change between two points on the line. This is also known as "rise over run". We use the formula for the slope, represented by the symbol \( m \): \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \((x_1, y_1)\) and \((x_2, y_2)\) are coordinates of the two points. A positive slope means the line is ascending from left to right, whereas a negative slope indicates it is descending. A zero slope implies a horizontal line, and undefined slope signifies a vertical line. Understanding how to calculate and interpret the slope is foundational in Analytic Geometry.

Parallel Lines

Parallel lines run at the same direction and will never intersect. For two lines to be parallel, their slopes must be equal. This means if you calculate the slopes of two lines and they turn out to be the same, the lines won't cross each other at any point.

• If the slope of the first line \( m_1 \) equals the slope of the second line \( m_2 \), then the lines are parallel.
• In simplified terms: \( m_1 = m_2 \)

Parallel lines are very essential in geometry, indicating equal travel paths without intersections, like rail tracks in most cases. Knowing this concept helps predict line behavior and understand geometrical relationships.

Perpendicular Lines

Perpendicular lines intersect at a right angle (90 degrees). For two lines to be perpendicular, the product of their slopes must be \(-1\). This means when you multiply the slopes of two perpendicular lines, the result is \(-1\).

• If the slope of the first line \( m_1 \) multiplied by the slope of the second line \( m_2 \) equals \(-1\), the lines are perpendicular: \( m_1 \times m_2 = -1 \).
• For instance, if one line has a slope of \(-\frac{1}{2}\) and the other line has a slope of \(2\), their product is \((-\frac{1}{2}) \times 2 = -1\).

Recognizing perpendicular lines is crucial in geometry as they form perfect angles, often necessary for construction and design applications.

Point-Slope Formula

The point-slope formula is used to find the equation of a line when you know the slope and one point on the line. It is a powerful tool in coordinate geometry for equations and graphical representations. The formula is set up as follows: \[ y - y_1 = m(x - x_1) \] Here, \((x_1, y_1)\) is the known point, and \( m \) is the slope of the line. This form is particularly useful because you can easily plug in a point and the slope to get the equation of a line.

• Allows you to write the line's equation quickly if you have the slope and any point on the line.
• Simplifies calculations and provides an immediate way to see how the line moves across the coordinate plane.

Mastering the point-slope formula aids in deeper comprehension of line equations and graph drawing, making it invaluable for geometry studies.