Problem 98 - College Algebra with Modeling and Visualization

Question

A rectangle is determined by the stated conditions. Find the slope-intercept form of the four lines that outline the rectangle. $$ \text { Vertices }(1,1),(5,1), \text { and }(5,5) $$

Step-by-Step Solution

Answer

The lines are: \(y = 1\), \(x = 5\), \(y = 5\), \(x = 1\).

Step 1: Identify the Vertices of the Rectangle

The given vertices are \((1,1)\), \((5,1)\), and \((5,5)\). One more vertex is needed to complete the rectangle. Since opposite vertices have the same y or x coordinate in rectangles, the missing vertex is \((1,5)\). So, the complete set of vertices is \((1,1), (5,1), (5,5), (1,5)\).

Step 2: Find the Equation of the First Line

The first line passes through \((1,1)\) and \((5,1)\). It is horizontal, so its equation is given by the constant y-coordinate of these points: \(y = 1\). In slope-intercept form, it is already \(y = mx + b\) with \(m = 0\) and \(b = 1\).

Step 3: Find the Equation of the Second Line

The second line passes through \((5,1)\) and \((5,5)\). It is vertical, so its x-coordinate is constant: \(x = 5\). Vertical lines don't have a slope-intercept form, but are described by such x-coordinate equations.

Step 4: Find the Equation of the Third Line

The third line passes through \((5,5)\) and \((1,5)\). It is horizontal, so its equation is given by the constant y-coordinate of these points: \(y = 5\). In slope-intercept form, it is \(y = mx + b\) with \(m = 0\) and \(b = 5\).

Step 5: Find the Equation of the Fourth Line

The fourth line passes through \((1,5)\) and \((1,1)\). It is vertical, so its x-coordinate is constant: \(x = 1\). Vertical lines are described by such x-coordinate equations, as they don't fit slope-intercept form.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to write the equation of a line. It is one of the most helpful formats in coordinate geometry, making it easy to identify a line’s slope and y-intercept quickly. The general equation is given by:\[ y = mx + b \]where:

• \(m\) represents the slope of the line.
• \(b\) is the y-intercept of the line, which is where the line crosses the y-axis.

In the case of horizontal lines, like those passing through the points (1,1) to (5,1) and (5,5) to (1,5), the slope \(m\) is zero, because there is no rise over the run. This simplifies our equation to \( y = b \). Similarly, for a vertical line, the concept of slope-intercept form doesn't apply because the slope is undefined.

Rectangle Vertices

In coordinate geometry, the corners of a rectangle are known as vertices. To form a rectangle, you need four vertices. These vertices are usually defined in a cyclic order, meaning you define one vertex and move around the rectangle until you come back to the starting point.Given vertices were \((1,1)\), \((5,1)\), and \((5,5)\). We can find the last vertex by recognizing the relationships:

• Opposite sides of a rectangle are equal.
• Instead of calculating the slope, we use consistent x or y values.

This tells us that if we have three vertices, we can use the equal coordinate property to find the fourth vertex. Completing the rectangle, we find that the fourth vertex is \((1,5)\). This creates four perfect lines that form a rectangle in the xy-plane.

Equation of a Line

The equation of a line serves as the foundation of analyzing linear relationships in coordinate geometry. This equation can take several forms, but the most common are the slope-intercept form and point-slope form.- Slope-Intercept Form: As mentioned, \(y = mx + b\) is the most straightforward form to understand a line's slope and where it intersects the y-axis.- Point-Slope Form: Useful when a line passes through a known point \((x_1, y_1)\) and has a slope \(m\). Its equation is:\[ y - y_1 = m(x - x_1) \]For horizontal and vertical lines in a rectangle, albeit slightly different rules apply:

• Horizontal lines, such as between points like (1,1) and (5,1), have equations \(y = b\).
• Vertical lines do not have a slope, hence the equation is simply \(x = a\), where \(a\) is the constant x-value.

These equations capture the geometry's linear properties and allow you to write simplified expressions for specific lines.

Horizontal and Vertical Lines

Horizontal and vertical lines are special cases in coordinate geometry, characterized by their unique orientation on a Cartesian plane.**Horizontal Lines**:- These lines run left-to-right and remain level in a consistent y-position.- Their equation is \(y = b\), indicating a constant y-coordinate.- No matter what the x-value is, the y stays the same.**Vertical Lines**:- They run up and down, maintaining a constant x-position.- Expressed as \(x = a\), meaning the x-coordinates do not change.- These lines have an undefined slope, which is why we can't express them in slope-intercept form.Knowing this distinction allows students to quickly identify and write equations for lines, enhancing their ability to work with geometric figures like rectangles.