Problem 63 - Algebra for College Students

Question

Solve each geometric figure problem. A trapezoid is a quadrilateral with one pair of parallel sides. Use slope to determine whether the points \((-3,2)\) \(-(-1,-1),(3,6),\) and \((6,4)\) are the vertices of a trapezoid.

Step-by-Step Solution

Answer

The points do not form a trapezoid as there are no parallel sides.

Step 1 - Calculate the slopes of each side

To determine if the figure is a trapezoid using the given points, calculate the slopes of the line segments formed by the given vertices. The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Calculate it for each side:

Step 2 - Slope of Line Segment \(-3,2\) to \(-1,-1\)

\(m_1 = \frac{-1 - 2}{-1 - (-3)} = \frac{-3}{2} = -\frac{3}{2}\)

Step 3 - Slope of Line Segment \(-1,-1\) to \(-3,6\)

\(m_2 = \frac{6 + 1}{3 - (-1)} = \frac{7}{4}\)

Step 4 - Slope of Line Segment \(-3,6\) to \(-6,4\)

\(m_3 = \frac{4 - 6}{6 - 3} = \frac{-2}{3} = -\frac{2}{3}\)

Step 5 - Slope of Line Segment \(-6,4\) to \(-3,2\)

\(m_4 = \frac{4 - 2}{6 - (-3)} = \frac{2}{9}\)

Step 6 - Compare slopes to determine parallel sides

After calculating the slopes, compare them to identify the parallel sides. Recall that a trapezoid has exactly one pair of parallel sides. The slopes found are: \(-\frac{3}{2}, \frac{7}{4}, -\frac{2}{3}, \frac{2}{9}\). None of these pairs are equal.

Step 7: Conclusion - Not a Trapezoid

Since no pair of slopes are equal, there are no parallel sides. Thus the given points do not form a trapezoid.

Key Concepts

slope formula

To solve geometric problems involving lines, the slope formula is a powerful tool. It helps us find the steepness of a line connecting two points. The formula to calculate the slope \(m\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Here, \(x_1, y_1\) and \(x_2, y_2\) are coordinates of the two points.
It's important to follow the steps carefully when applying the slope formula. Always subtract the \(y\)-coordinates (rise) over the \(x\)-coordinates (run).
Using this method, you can determine the slopes of all segments of a geometric figure. Comparing these slopes enables you to find parallel sides and solve various problems, like identifying a trapezoid.

geometric figures

Understanding geometric figures is crucial in solving algebraic problems. A geometric figure is a space defined by points, lines, curves, and surfaces.

• To identify different figures, observe their sides and angles.
• Quadrilaterals, for example, have four sides.

Geometric figures in algebra often require calculating lengths, areas, and determining relationships like parallelism.
In this problem, our goal is to verify if the given points form a trapezoid. We rely on the properties of geometric figures, particularly the properties of trapezoids. Using the right methods and definitions simplifies solving such problems.

parallel lines

Parallel lines are lines that never meet, no matter how far they extend. Their slopes are equal, which is the key property we use to identify them.
If two line segments have the same slope, they're parallel. This concept is crucial when determining the type of quadrilateral formed by given points.

• A trapezoid has at least one pair of parallel sides.
• If no pair is parallel, the figure isn't a trapezoid.

In our exercise, none of the calculated slopes are equal. Thus, no sides are parallel, indicating the points do not form a trapezoid.

quadrilaterals

Quadrilaterals are four-sided polygons. They can be classified into sub-categories based on side and angle properties.

• Squares and rectangles have all sides and all angles equal, respectively.
• Parallelograms have opposite sides parallel and equal.
• Trapezoids have exactly one pair of parallel sides.

When given a set of points, identifying the type of quadrilateral involves calculating slopes and lengths.
Comparing properties like parallelism and symmetry helps determine the specific type. In the outlined exercise, slope calculations showed no parallel sides, proving that the points don't form a trapezoid.