Problem 8 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 7-10: Use slopes to show that the points are vertices of the specified polygon. $$ A(2,3), B(5,-1), C(0,-6), D(-6,2) ; \quad \text { trapezoid } $$

Step-by-Step Solution

Answer

The points form a trapezoid because lines AB and CD are parallel.

Step 1: Find the Slope of Line AB

To find the slope of the line through points \(A(2,3)\) and \(B(5,-1)\), use the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For points A and B, substitute \(x_1 = 2, y_1 = 3, x_2 = 5, y_2 = -1\). Thus, the slope is: \[ m_{AB} = \frac{-1 - 3}{5 - 2} = \frac{-4}{3}. \]

Step 2: Find the Slope of Line CD

Next, calculate the slope of the line through points \(C(0,-6)\) and \(D(-6,2)\). Again, use the slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substitute \(x_1 = 0, y_1 = -6, x_2 = -6, y_2 = 2\). The slope is: \[ m_{CD} = \frac{2 + 6}{-6 - 0} = \frac{8}{-6} = \frac{-4}{3}. \]

Step 3: Verify Parallelism of AB and CD

Since the slopes of lines AB and CD, \(m_{AB}\) and \(m_{CD}\), are both \(\frac{-4}{3}\), these lines are parallel. For a trapezoid, at least one pair of opposite sides must be parallel, confirming AB and CD suffice for this condition.

Step 4: Find the Slope of Line BC

Calculate the slope of the line through points B(5,-1) and C(0,-6). Using the slope formula, substitute \(x_1 = 5, y_1 = -1, x_2 = 0, y_2 = -6\). The slope is: \[ m_{BC} = \frac{-6 + 1}{0 - 5} = \frac{-5}{-5} = 1. \]

Step 5: Find the Slope of Line DA

Finally, find the slope of the line through points D(-6,2) and A(2,3). Again, use the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substitute \(x_1 = -6, y_1 = 2, x_2 = 2, y_2 = 3\). The slope is: \[ m_{DA} = \frac{3 - 2}{2 + 6} = \frac{1}{8}. \]

Step 6: Confirm No Additional Parallelism

The slopes of BC and DA, \(1\) and \(\frac{1}{8}\), are different, indicating that lines BC and DA are not parallel. Thus, we have only one pair of parallel sides, confirming the figure satisfies the conditions of a trapezoid.

Key Concepts

Slope Formula

The slope formula is a fundamental concept in coordinate geometry. It helps us determine the steepness of a line connecting two points. Imagine a ski slope; some are steep and some are not. Similarly, in math, each line has a steepness or slope.
To calculate the slope of a line through two points, we use: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here,

• \( m \) stands for the slope.
• \( x_1, y_1 \) are the coordinates of the first point.
• \( x_2, y_2 \) are the coordinates of the second point.

This formula gives us the rate at which \( y \) changes for every unit increase in \( x \). If you plug in those values, you'll easily get the slope of the line.
For example, the slope for points \( A(2, 3) \) and \( B(5, -1) \) is \( \frac{-4}{3} \), reflecting how the surface of this imaginary slope descends.

Parallel Lines

Parallel lines are fascinating in that they never touch. Imagine train tracks that run next to each other for miles. In geometry, for lines to be parallel, they must have identical slopes.
This means they incline or decline precisely the same amount. When working with coordinates and slopes, if the slopes of two lines are equal, then those lines are parallel.
To check for parallelism using the slope formula, you compare slopes calculated from the two line segments.
In our exercise, lines \( AB \) and \( CD \) both have a slope of \( \frac{-4}{3} \). Since they share the same slope, these lines are parallel, much like hypothetical parallel train tracks on your geometry plane.
This single characteristic is essential in recognizing the structure of a trapezoid.

Coordinate Geometry

Coordinate geometry, often called analytic geometry, combines algebra and geometry. It's about telling the story of shapes using a coordinate plane, which is essentially a grid.
Each point on this grid is noted by pairs of numbers specifiying their location:

• The first number (x-coordinate) shows the horizontal position.
• The second number (y-coordinate) shows the vertical position.

By working with these coordinates, we can explore relationships between lines and shapes, like understanding how they orient or interact.
In the context of our trapezoid exercise, the points \( A, B, C, \) and \( D \) on the plane determine the vertices of this four-sided figure. We use these to calculate slopes, check parallelism, and confirm geometric properties.
Using coordinate geometry to show how lines and points fit together really helps in visualizing and proving the properties of shapes like trapezoids.