Problem 7 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

The points form a parallelogram because both pairs of opposite sides are parallel.

Step 1: Calculate the slope of AB

The slope of a line is calculated as \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For the line AB, use the coordinates of points A and B, which are \((-3,1)\) and \((5,3)\). So, the slope of AB is \( m_{AB} = \frac{3 - 1}{5 - (-3)} = \frac{2}{8} = \frac{1}{4} \).

Step 2: Calculate the slope of CD

Using the slope formula for points C and D, which are \((3,0)\) and \((-5,-2)\), find the slope of CD: \( m_{CD} = \frac{0 - (-2)}{3 - (-5)} = \frac{2}{8} = \frac{1}{4} \).

Step 3: Verify Parallelism Between AB and CD

Since \( m_{AB} = \frac{1}{4} \) and \( m_{CD} = \frac{1}{4} \), AB and CD are parallel lines because they have the same slope.

Step 4: Calculate the slope of BC

Using the slope formula, find the slope of BC using points B \((5,3)\) and C \((3,0)\): \( m_{BC} = \frac{0 - 3}{3 - 5} = \frac{-3}{-2} = \frac{3}{2} \).

Step 5: Calculate the slope of DA

Now, calculate the slope of DA using points D \((-5,-2)\) and A \((-3,1)\): \( m_{DA} = \frac{1 - (-2)}{-3 - (-5)} = \frac{3}{2} \).

Step 6: Verify Parallelism Between BC and DA

Since \( m_{BC} = \frac{3}{2} \) and \( m_{DA} = \frac{3}{2} \), the lines BC and DA are parallel as they share the same slope.

Step 7: Conclude the Shape as a Parallelogram

Both pairs of opposite sides AB and CD, BC and DA are parallel. A quadrilateral is a parallelogram if both pairs of opposite sides are parallel, thus confirming that the figure is a parallelogram.

Key Concepts

Slope of a line

The slope of a line is a measure of its steepness and direction. To determine the slope, you use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where

• \( (x_1, y_1) \) is the coordinate of the first point,
• \( (x_2, y_2) \) is the coordinate of the second point.

This simple formula indicates how much the line rises or falls with respect to each unit it moves horizontally.
A positive slope indicates that the line ascends from left to right. Conversely, a negative slope means it descends. When the slope is zero, the line is horizontal, and if the slope is undefined, the line is vertical.

Parallel lines

When two lines have the same slope, they never intersect and are considered parallel. In the context of our exercise with points A, B, C, and D, we found the slopes of certain lines to determine parallelism.
For example:

• The slope of AB is \( \frac{1}{4} \).
• The slope of CD is identical at \( \frac{1}{4} \); therefore, AB and CD are parallel.
• Similarly, lines BC and DA had slopes of \( \frac{3}{2} \), making them parallel too.

Parallel lines are significant because they can help us identify the shape and properties of polygons. In a parallelogram, opposite sides are always parallel. Recognizing this can simplify complex geometric problems.

Quadrilateral

A quadrilateral is a four-sided polygon with four vertices. The main types of quadrilaterals include squares, rectangles, trapezoids, and parallelograms.
The properties of a quadrilateral can be determined by examining its sides and angles. How the sides relate to each other, especially concerning their length and parallelism, can define its type. For this exercise, we were tasked with proving that our shape, with vertices at A, B, C, and D, is a parallelogram.

• By verifying that opposite sides are parallel (AB \(\parallel\) CD and BC \(\parallel\) DA), we proved that it is a parallelogram.
• Knowing that a parallelogram's opposite sides are equal in length also helps, although not needed for slope calculations in this case.

Understanding these properties allows for powerful deductions about the figure's measurements and angles, streamlining problem-solving in geometry.