Problem 53 - College Algebra with Corequisite Support

Question

Given that the following coordinates are the vertices of a rectangle, prove that this truly is a rectangle by showing the slopes of the sides that meet are perpendicular. (-1,1),(2,0),(3,3) and (0,4).

Step-by-Step Solution

Answer

The sides AB and BC, as well as CD and DA, are perpendicular, confirming a rectangle.

Step 1: Plot the Points

Visualize the problem by plotting the given points on a coordinate plane. The vertices are A(-1,1), B(2,0), C(3,3), and D(0,4). This will help in understanding the position of each point relative to each other.

Step 2: Determine the Slope of AB

Calculate the slope of segment AB using the slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For points A(-1,1) and B(2,0):\[ m_{AB} = \frac{0 - 1}{2 - (-1)} = \frac{-1}{3} \]

Step 3: Determine the Slope of BC

Calculate the slope of segment BC. For points B(2,0) and C(3,3):\[ m_{BC} = \frac{3 - 0}{3 - 2} = 3 \]

Step 4: Check the Perpendicular Relationship of AB and BC

To prove the sides are perpendicular, the product of their slopes should be -1. For sides AB (-1/3) and BC (3), multiply the slopes:\[ m_{AB} \times m_{BC} = \frac{-1}{3} \times 3 = -1 \] Thus, AB and BC are perpendicular.

Step 5: Determine the Slope of CD

Calculate the slope of segment CD. For points C(3,3) and D(0,4):\[ m_{CD} = \frac{4 - 3}{0 - 3} = -\frac{1}{3} \]

Step 6: Determine the Slope of DA

Calculate the slope of segment DA. For points D(0,4) and A(-1,1):\[ m_{DA} = \frac{1 - 4}{-1 - 0} = 3 \]

Step 7: Check the Perpendicular Relationship of CD and DA

Check if the product of slopes of CD and DA is -1:\[ m_{CD} \times m_{DA} = -\frac{1}{3} \times 3 = -1 \]Therefore, CD and DA are perpendicular.

Key Concepts

Slope of a Line

In analytic geometry, the slope of a line is a measure of the steepness or incline of the line. It tells us how much the line rises vertically for every unit it moves horizontally.
The slope is commonly represented by the letter "m" and is calculated using the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This formula subtracts the y-values and the x-values of two points on the line. It gives you the change in y divided by the change in x.For example, let's find the slope between two points A(-1,1) and B(2,0). Plug the values into the slope formula:

• Difference in y-values = 0 - 1 = -1
• Difference in x-values = 2 - (-1) = 3

So, the slope \( m_{AB} = \frac{-1}{3} \). That means for every 3 units you travel horizontally, the line descends by 1 unit.

Perpendicular Lines

Perpendicular lines are an important concept in geometry. When two lines are perpendicular, they intersect at a right angle (90 degrees).
This right angle forms the basis for many geometric shapes and properties.The relationship between the slopes of two perpendicular lines is interesting. Two lines are perpendicular if the product of their slopes is -1.

• If you have a line with a slope of \( m \), a perpendicular line will have a slope of \( -\frac{1}{m} \).

For example, let's examine this with the perpendicular lines AB and BC. We have:

• Slope of AB: \( m_{AB} = \frac{-1}{3} \)
• Slope of BC: \( m_{BC} = 3 \)

Calculate the product of these slopes: \( m_{AB} \times m_{BC} = \frac{-1}{3} \times 3 = -1 \). This confirms they are perpendicular.

Rectangle Properties

A rectangle is a four-sided polygon that has a few distinct properties, making it a special type of quadrilateral. Each corner of a rectangle forms a right angle.
This means all interior angles measure precisely 90 degrees.The most recognizable property of a rectangle is that opposite sides are equal and parallel. Another key feature is that two adjacent sides of a rectangle are perpendicular.In the given exercise, we determined the sides AB and BC to be perpendicular, forming a right angle. Similarly, sides CD and DA were shown to be perpendicular as well.

• Slope of CD: \( m_{CD} = -\frac{1}{3} \)
• Slope of DA: \( m_{DA} = 3 \)
• Product: \( m_{CD} \times m_{DA} = -1 \)

These perpendicular relationships between pairs of adjacent sides demonstrate the necessary conditions to confirm the shape is indeed a rectangle.