Problem 36 - Introductory Algebra for College Students

Question

In Exercises \(27-30\), determine whether the lines through each pair of points are perpendicular. $$(-2,-15) \text { and }(0,-3) ;(-12,6) \text { and }(6,3)$$

Step-by-Step Solution

Answer

Yes, the lines are perpendicular.

Step 1: Calculate the slope of the first line

To find the slope of the first line, we plug the points \((-2,-15)\) and \(0,-3)\) into our slope formula. The slope \(m_1\) of the first line is calculated as follows: \(m_1 = (-3 --15) / (0 --2) = 12 / 2 = 6\)

Step 2: Calculate the slope of the second line

Next, we calculate the slope of the second line. Using the points \((-12,6)\) and \(6,3)\), we find the slope \(m_2\) as follows: \(m_2 = (3 - 6) / (6 --12) = -3 / 18 = -1/6\)

Step 3: Check if the lines are perpendicular

Two lines are perpendicular if the product of their slopes is -1. We check this by multiplying the slopes of the lines we calculated in the previous steps. If \(m_1 * m_2 = -1\), then the lines are perpendicular. For our example: \(m_1 * m_2 = 6 * -1/6 = -1\)

Key Concepts

Slope Calculation

The slope of a line is a measure of its steepness and is a crucial concept in coordinate geometry. When you're given two points on a line, you can calculate the slope by finding the ratio of the vertical change (rise) to the horizontal change (run) between these points.

For any two points \( (x_1, y_1) \) and \( (x_2, y_2) \), the slope \( m \) can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \.
\] For instance, given the points \( (-2, -15) \) and \( (0, -3) \), the slope \( m \) would be \( 6 \) as calculated in the step by step solution.

Slope Formula

The slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \) is the standardized way to find the slope of a line passing through two specific points. It’s important to apply the formula correctly and maintain consistency in using the coordinates for \( y_2 - y_1 \) and \( x_2 - x_1 \) - mixing these up can lead to the wrong value for the slope!

Applying this to our given points for the second line, \( (-12,6) \) and \( (6,3) \) gives us the slope \( m \) of \( -\frac{1}{6} \) as shown in the solution provided.

Determining Perpendicularity of Lines

When we need to determine if two lines are perpendicular, we exploit the relationship between their slopes. Perpendicular lines in a coordinate plane have slopes that are negative reciprocals of each other, meaning that the product of their slopes is \( -1 \).

If we look at our example, the slopes \( m_1 \) and \( m_2 \) of the lines are \( 6 \) and \( -\frac{1}{6} \) respectively. To check if these lines are perpendicular, we calculate the product \( m_1 \times m_2 = 6 \times -\frac{1}{6} \) which indeed equals \( -1 \) confirming their perpendicularity.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, deals with the connection between algebra and geometry through the use of a coordinate system. It allows us to solve geometric problems by means of algebraic equations. In this field, concepts like the slope of lines, distance between points, and characteristics of figures like triangles and circles are analyzed within the xy-plane.

In the context of our exercise, coordinate geometry concepts are applied to first calculate slopes and then determine the relationship between those lines – in this case, checking if they are perpendicular through slope analysis.