Problem 50 - Calculus Single Variable

Question

Is the line through points (-3,8) and (2,23) parallel to the line through points (-4,6) and (3,28)\(?\) Is it perpendicular to the line through points (-2,4) and (7,1)\(?\)

Step-by-Step Solution

Answer

The lines are not parallel but are perpendicular to each other.

Step 1: Calculate the slope of the first line

To find the slope of the line through points (-3, 8) and (2, 23), use the slope formula: \( m_1 = \frac{y_2 - y_1}{x_2 - x_1} \). Substitute the coordinates to get \( m_1 = \frac{23 - 8}{2 + 3} = \frac{15}{5} = 3 \).

Step 2: Calculate the slope of the second line

For the line through points (-4, 6) and (3, 28), again use the slope formula: \( m_2 = \frac{28 - 6}{3 + 4} = \frac{22}{7} \).

Step 3: Check if the lines are parallel

Two lines are parallel if their slopes are equal. Since \( m_1 = 3 \) and \( m_2 = \frac{22}{7} \), and these are not equal, the lines are not parallel.

Step 4: Calculate the slope of the third line

For the line through points (-2, 4) and (7, 1), use the slope formula: \( m_3 = \frac{1 - 4}{7 + 2} = \frac{-3}{9} = -\frac{1}{3} \).

Step 5: Check if the lines are perpendicular

Two lines are perpendicular if the product of their slopes is -1. Calculate \( m_1 \cdot m_3 = 3 \cdot -\frac{1}{3} = -1 \). Since the product is -1, the lines are perpendicular.

Key Concepts

Parallel Lines

Parallel lines are fascinating to study in the realm of geometry and algebra. When we say two lines are parallel, we mean that no matter how far you extend them, they will never meet. This unique property of parallel lines is a result of their equal slopes. In simpler terms, slope is a measure of how much a line rises vertically compared to how far it runs horizontally.

To determine if two lines are parallel, you compare their slopes. Recall, the slope (\( m \)) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as \( m = \frac{y_2 - y_1}{x_2 - x_1} \). If two lines have the same slope, \( m_1 = m_2 \), they are parallel.

In our exercise, we found the slope of the first line to be 3, but the second line's slope is \(\frac{22}{7}\). Since these slopes aren't the same, the lines through the given points are not parallel.

Perpendicular Lines

Perpendicular lines create a unique intersection, forming right angles. This intersection is crucial in geometry, offering a distinctive property. Such lines have slopes that multiply to give -1. This characteristic is crucial for determining perpendicularly, using the slope product method.

To confirm if lines are perpendicular, calculate each line's slope. Let's say one line's slope is \( m_1 \) and another's is \( m_3 \). For these lines to be perpendicular, \( m_1 \cdot m_3 = -1 \).

In our exercise, the slope of the first line is 3, while that of another line is \(-\frac{1}{3}\). Computing their product, \( 3 \cdot -\frac{1}{3} = -1 \). As a result, these lines are perpendicular owing to their slope relationship.

Equation of a Line

Understanding the equation of a line is pivotal in math. It allows you to describe a line's path on a graph. The most common form of a line's equation is the slope-intercept form: \( y = mx + b \). Here, \( m \) represents the slope, and \( b \) is the y-intercept, where the line crosses the y-axis.

To craft an equation, you need the slope and a point on the line. With a known point \((x_1, y_1)\) and slope \( m \), use the point-slope formula:

• \( y - y_1 = m(x - x_1) \)
• Rearrange to find the slope-intercept form

Consider a line through points (-3, 8) and (2, 23). We know the slope \( m \) is 3. Using the point (-3, 8), plug in the values:

• \( y - 8 = 3(x + 3) \)
• Simplifying gives \( y = 3x + 17 \)

The final equation represents that particular line on the graph.