Problem 57 - Precalculus Mathematics for Calculus

Question

Use slopes to show that \(A(1,1), B(7,4), C(5,10),\) and \(D(-1,7)\) are vertices of a parallelogram.

Step-by-Step Solution

Answer

The points are vertices of a parallelogram as opposite sides are parallel.

Step 1: Calculate Slope of AB

The slope of a line between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Calculate the slope of line segment AB where \(A(1,1)\) and \(B(7,4)\). Thus, the slope \(m_{AB} = \frac{4 - 1}{7 - 1} = \frac{3}{6} = \frac{1}{2}.\)

Step 2: Calculate Slope of CD

Use the slope formula for points \(C(5,10)\) and \(D(-1,7)\). The slope \(m_{CD} = \frac{7 - 10}{-1 - 5} = \frac{-3}{-6} = \frac{1}{2}.\)

Step 3: Compare Slopes of AB and CD

Compare the slopes obtained from Steps 1 and 2. Since \(m_{AB} = \frac{1}{2}\) and \(m_{CD} = \frac{1}{2}\), lines AB and CD are parallel.

Step 4: Calculate Slope of BC

Now calculate the slope of line segment BC using points \(B(7,4)\) and \(C(5,10)\). The slope \(m_{BC} = \frac{10 - 4}{5 - 7} = \frac{6}{-2} = -3.\)

Step 5: Calculate Slope of AD

Calculate the slope of AD using points \(A(1,1)\) and \(D(-1,7)\). The slope \(m_{AD} = \frac{7 - 1}{-1 - 1} = \frac{6}{-2} = -3.\)

Step 6: Compare Slopes of BC and AD

Compare slopes from Steps 4 and 5. Since \(m_{BC} = -3\) and \(m_{AD} = -3\), lines BC and AD are parallel.

Step 7: Conclusion: Verify Parallelogram Condition

Since opposite sides AB \(\parallel\) CD and BC \(\parallel\) AD, the points form a parallelogram by definition.

Key Concepts

Slope

When studying lines, understanding the concept of slope is crucial. The slope of a line is a measure of its steepness, often calculated as the 'rise' over the 'run,' which translates to the change in the y-coordinates divided by the change in the x-coordinates. This is mathematically expressed using the formula:

• For two points \( (x_1, y_1) \) and \( (x_2, y_2) \), the slope \( m \) is given by \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

A positive slope means the line ascends as it moves from left to right, while a negative slope means it descends. A slope of zero indicates a horizontal line, which is entirely level.
In parallel lines, like in the given exercise, the slopes are equal, indicating they run in the same direction without ever intersecting.

Parallel Lines

Parallel lines are lines in a plane that never meet; they are always the same distance apart. In terms of coordinates and slope, two lines are parallel when they have identical slopes. This quality is helpful in identifying shapes like parallelograms.
In our exercise, lines AB and CD are shown to have the same slope (\( \frac{1}{2} \)), as well as lines BC and AD with a slope of \(-3\).

• This parallelism between AB and CD, and BC and AD, is essential to proving that the figure outlined by these points is a parallelogram.

Vertices of a Shape

Vertices are the corner points of a shape where two or more edges meet. In coordinate geometry, each vertex is represented by a set of coordinates. For our shape, we have four vertices: A(1,1), B(7,4), C(5,10) and D(-1,7).

• Identifying these vertices is the first step in determining what kind of shape they form.
• It's essential to understand their connections, as this helps in determining the shape of the polygon they form.

In our problem, the relationship between these vertices is analyzed by comparing the slopes of the lines connecting them, leading us to conclude the shape is a parallelogram.

Coordinate Geometry

Coordinate geometry, or analytic geometry, allows us to study geometry using a coordinate system, combining algebra and geometry. This system makes it possible to describe geometric figures analytically and solve geometric problems numerically.

• Using coordinates \( (x, y) \), we can pinpoint exact locations in the plane, which is particularly useful in identifying shapes like parallelograms.
• By calculating slopes, we can derive relationships between points and use these relationships to confirm the nature of the geometric figure formed.

Coordinate geometry provides the tools needed to verify the problem's claim: by proving parallelism through slope and visualizing the shape formed by the given vertices, we confirm it is indeed a parallelogram.