Problem 53 - College Algebra

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 quadrilateral is a rectangle because its consecutive sides are perpendicular.

Step 1: Plot the Points

First, let's plot the points \((-1,1), (2,0), (3,3),\) and \((0,4)\). Identify the possible sides. Use the coordinates to visualize the quadrilateral formed by these points.

Step 2: Calculate Slopes of Sides

Calculate the slope of each side by selecting pairs of points: A = \((-1,1)\), B = \((2,0)\), C = \((3,3)\), and D = \((0,4)\).- For side AB: Use the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). \[ m_{AB} = \frac{0 - 1}{2 - (-1)} = \frac{-1}{3} \]- For side BC: \[ m_{BC} = \frac{3 - 0}{3 - 2} = 3 \]- For side CD: \[ m_{CD} = \frac{4 - 3}{0 - 3} = \frac{1}{-3} = -\frac{1}{3} \]- For side DA: \[ m_{DA} = \frac{1 - 4}{-1 - 0} = \frac{-3}{-1} = 3 \]

Step 3: Check Perpendicularity

To prove that the quadrilateral is a rectangle, opposite sides must be perpendicular. Two lines are perpendicular if the product of their slopes is -1.- Check if sides AB and BC are perpendicular: \[ m_{AB} \times m_{BC} = \left(-\frac{1}{3}\right) \times 3 = -1 \] Thus, AB is perpendicular to BC.- Check if sides CD and DA are perpendicular: \[ m_{CD} \times m_{DA} = \left(-\frac{1}{3}\right) \times 3 = -1 \] Thus, CD is perpendicular to DA.

Step 4: Conclude It is a Rectangle

Since each pair of consecutive sides is perpendicular, the quadrilateral formed by the points is a rectangle. We verified perpendicularity using the calculated slopes.

Key Concepts

Coordinate Geometry

Coordinate geometry is a branch of geometry where we use a coordinate system to describe the positions of points or shapes in space. Here, the Cartesian coordinate system is used, allowing us to locate each vertex of a quadrilateral on a plane using pairs of numerical values called coordinates,

• The x-coordinate represents the horizontal position.
• The y-coordinate indicates the vertical position.

By plotting the given vertices' coordinates, we visually and mathematically identify the shape of the quadrilateral. This process establishes a foundation for further analytical calculations like determining slopes of sides. By positioning our points accurately, we can apply formulas to examine relationships between these points that reveal significant properties of quadrilaterals.

Slope Calculation

In coordinate geometry, slope calculation is a critical method used to determine the steepness or incline of a line connecting two points. The slope, denoted as \(m\), answers how much y changes with respect to x.The formula for calculating the slope between two points, \((x_1, y_1)\) and \((x_2, y_2)\), is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

• For line AB connecting \((-1,1)\) and \((2,0)\): \[ m_{AB} = \frac{0 - 1}{2 - (-1)} = \frac{-1}{3} \]
• For line BC connecting \((2,0)\) and \((3,3)\): \[ m_{BC} = \frac{3 - 0}{3 - 2} = 3 \]
• For line CD connecting \((3,3)\) and \((0,4)\): \[ m_{CD} = \frac{4 - 3}{0 - 3} = -\frac{1}{3} \]
• For line DA connecting \((0,4)\) and \((-1,1)\): \[ m_{DA} = \frac{1 - 4}{-1 - 0} = 3 \]

Understanding slope calculations helps further analyze properties, like perpendicularity, which are vital for proving shapes such as rectangles.

Perpendicular Lines

Two lines are perpendicular if they intersect at a right angle (90 degrees). In coordinate geometry, we ascertain this by examining their slopes — specifically, the slopes of these two lines multiply to produce -1.In this exercise, the slopes of opposite sides were calculated:

• For lines AB and BC: \(m_{AB} = -\frac{1}{3}\), \(m_{BC} = 3\); and their product: \[ m_{AB} \times m_{BC} = -\frac{1}{3} \times 3 = -1 \]
• For lines CD and DA: \(m_{CD} = -\frac{1}{3}\), \(m_{DA} = 3\); and their product: \[ m_{CD} \times m_{DA} = -\frac{1}{3} \times 3 = -1 \]

This calculation forms the basis for determining whether consecutive sides of a quadrilateral intersect perpendicularly, an essential aspect when confirming shapes like rectangles.

Quadrilateral Properties

Quadrilaterals are four-sided polygons, with rectangles being a specific type. A key property of rectangles is that they have all right angles (90 degrees) between adjacent sides. To confirm a quadrilateral is a rectangle using coordinate geometry, we demonstrate:

• Each pair of consecutive sides is perpendicular, as proven by their slopes multiplying to -1.
• The quadrilateral still must be closed, with points that connect into a four-sided shape without crossing over.

Knowing these properties allows us to analytically prove quadrilaterals as rectangles through slope calculations, confirming the presence of right angles between sides. This approach offers a precise and mathematical method to validate geometric figures.