Problem 74 - Algebra for College Students

Question

What is the slope of a line that is perpendicular to the line through \((3.27,-1.46)\) and \((-5.48,3.61) ?\)

Step-by-Step Solution

Answer

The slope of the perpendicular line is approximately 1.72.

Step 1 - Determine the slope of the given line

Use the slope formula \(-1 m = \frac{y_2 - y_1}{x_2 - x_1}\) to calculate the slope of the line through the points \((3.27, -1.46)\) and \((-5.48, 3.61)\). Here, \((x_1, y_1) = (3.27, -1.46)\) and \((x_2, y_2) = (-5.48, 3.61)\). Substitute these values into the formula to find the slope.

Step 2 - Calculate the slope

Substitute the points into the slope formula: \[ m = \frac{3.61 - (-1.46)}{-5.48 - 3.27} = \frac{3.61 + 1.46}{-5.48 - 3.27} = \frac{5.07}{-8.75} = -0.58\] So, the slope of the given line is \(-0.58\).

Step 3 - Determine the slope of the perpendicular line

The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. If the original slope is \(m = -0.58\), then the slope of the perpendicular line is \(m' = \frac{-1}{-0.58} = \frac{1}{0.58}\).

Step 4 - Simplify the slope of the perpendicular line

Simplify the slope: \[\frac{1}{0.58} \approx 1.72\] Therefore, the slope of the perpendicular line is approximately \1.72\.

Key Concepts

Slope Calculation

Understanding the slope of a line is essential in coordinate geometry. The slope tells us how steep a line is. It is a measure of the vertical change per unit of horizontal change between two points.
The formula for calculating the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This formula finds the 'rise' (change in y) over the 'run' (change in x). For example, using points \( (3.27, -1.46) \) and \( (-5.48, 3.61) \), we substitute into the formula:
\[ m = \frac{3.61 - (-1.46)}{-5.48 - 3.27} = \frac{3.61 + 1.46}{-5.48 - 3.27} = \frac{5.07}{-8.75} = -0.58 \] Here, the slope \( m \) of the line is \(-0.58\). It's important to remember the calculated slope helps us understand the direction and steepness of the line.

Negative Reciprocal

When studying perpendicular lines, a key concept is the negative reciprocal of the slope. Two lines are perpendicular if the product of their slopes is -1.

To find the slope of a line perpendicular to another, you use the negative reciprocal. This means:

• First, take the reciprocal (flip the fraction) of the original slope.
• Then, change the sign from positive to negative or negative to positive.

For a line with slope \(-0.58\), the negative reciprocal is found as follows:

• Reciprocal of \(-0.58\) is \(\frac{1}{-0.58}\).
• The negative sign converts it to positive, yielding approximately \(\frac{1}{0.58} = 1.72\).

Thus, the slope of the perpendicular line is \(1.72\). This concept ensures we can accurately match the geometry required for right angles in coordinate systems.

Coordinate Geometry

Coordinate geometry, or analytic geometry, involves algebraic operations to solve geometric problems. Knowing how to work with points, lines, and slopes is crucial.

To start, understand the coordinate plane where any point is represented as \((x, y)\). By identifying these points, you can use formulas like the slope formula to analyze relationships between points and lines. For example, determining slopes helps reveal:

• If lines are parallel (same slopes).
• If lines are perpendicular (negative reciprocal slopes).

Working through these calculations clarifies complex geometric shapes and their properties. For our specific exercise, plotting points \((3.27, -1.46)\) and \((-5.48, 3.61)\) and using the slope formula reveals the nature of the line connecting these points.

Coordinate geometry integrates algebra and geometry, allowing precise descriptions of spatial relationships. This integration aids in solving real-world problems by creating accurate geometric models.