Problem 61 - Algebra for College Students

Question

Solve each geometric figure problem. If the opposite sides of a quadrilateral are parallel, then it is a parallelogram. Use slope to determine whether the points \((-6,1),(-2,-1),(0,3),\) and \((4,1)\) are the vertices of a parallelogram.

Step-by-Step Solution

Answer

The points do not form a parallelogram because the opposite sides are not all parallel.

Step 1: Calculate the slope of each side

To determine if the opposite sides of the quadrilateral are parallel, calculate the slopes of the lines formed by the given points. The formula for the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is: \[ \text{Slope} = \frac{y_2 - y_1}{x_2 - x_1} \].

Step 2: Calculate the slope of sides AB and CD

Points A(-6,1) and B(-2,-1): \[ \text{Slope}_{AB} = \frac{-1 - 1}{-2 - (-6)} = \frac{-2}{4} = -\frac{1}{2} \]. Points C(0,3) and D(4,1): \[ \text{Slope}_{CD} = \frac{1 - 3}{4 - 0} = \frac{-2}{4} = -\frac{1}{2} \].

Step 3: Calculate the slope of sides BC and DA

Points B(-2,-1) and C(0,3): \[ \text{Slope}_{BC} = \frac{3 - (-1)}{0 - (-2)} = \frac{4}{2} = 2 \]. Points D(4,1) and A(-6,1): \[ \text{Slope}_{DA} = \frac{1 - 1}{-6 - 4} = \frac{0}{-10} = 0 \].

Step 4: Compare opposite side slopes

Compare the slopes calculated to see if the opposite sides are parallel: \[ \text{Slope}_{AB} = -\frac{1}{2} \] and \[ \text{Slope}_{CD} = -\frac{1}{2} \] (parallel). \[ \text{Slope}_{BC} = 2 \] and \[ \text{Slope}_{DA} = 0 \] (not parallel). Since the opposite sides \(BC \) and \(DA \) are not parallel, this quadrilateral is not a parallelogram.

Key Concepts

Slopes of Lines

In geometry, understanding slopes is crucial. The slope of a line quantifies its steepness. It's a measure that points to how inclined or flat a line is. Slope is determined by the vertical change ('rise') over the horizontal change ('run') between two points on the line. The slope formula is expressed as: \[ \text{Slope} = \frac{y_2 - y_1}{x_2 - x_1} \]Here,

• \((x_1, y_1)\) and \((x_2, y_2)\) are coordinates of two points on the line.
• When the slope is positive, the line ascends from left to right.
• When negative, it descends from left to right.

Parallel lines share the same slope, while perpendicular lines have slopes that are negative reciprocals of one another.

Parallelogram Criteria

A quadrilateral is a parallelogram if both pairs of its opposite sides are parallel. This means:

• Each pair of opposite sides must have the same slope.
• Knowing the slopes of lines is pivotal to verifying the shape.

Using the given points \((-6,1),(-2,-1),(0,3),(4,1)\), we calculated the slopes:

• Slope of AB: \[-\frac{1}{2}\]
• Slope of CD: \[-\frac{1}{2}\] (parallel to AB)
• Slope of BC: \[2\]
• Slope of DA: \[0\] (not parallel to BC)

Hence, only one pair is parallel (AB and CD). For it to be a parallelogram, both sets of opposite sides must be parallel. Therefore, this quadrilateral isn't a parallelogram.

Geometric Proofs

Geometric proofs validate the properties of shapes using logical deductions and calculations. In this example, we used slopes to show the quadrilateral isn’t a parallelogram. Here’s the step-by-step approach: 1. **Identify Coordinates**: Note down the coordinates of the vertices.
For instance, the given points are \((-6,1),(-2,-1),(0,3),(4,1)\).
2. **Apply Slope Formula**: Use the slope formula for each side.
\[ \text{Slope} = \frac{y_2 - y_1}{x_2 - x_1} \]
3. **Determine Parallelism**: Compare slopes of opposite sides. Parallel lines have equal slopes. Non-equal slopes mean non-parallel lines. If both pairs of opposite sides are parallel, it confirms a parallelogram.
4. **Conclude**: Given the findings, deduce the geometric property. In this case, our finding concluded that it isn't a parallelogram. Practicing these steps sharpens problem-solving and proof skills in geometry.