Problem 77 - Algebra for College Students

Question

A rhombus is a quadrilateral with four equal sides. Draw a rhombus with vertices \((-3,-1),(0,3),(2,-1)\) and \((5,3) .\) Find the slopes of the diagonals of the rhombus. What can you conclude about the diagonals of this rhombus?

Step-by-Step Solution

Answer

The slopes of the diagonals are \(\frac{1}{2}\) and \(-2\). The diagonals are perpendicular.

Step 1: Identify the vertices

List the coordinates of the rhombus vertices provided in the problem. The vertices are \((-3, -1)\), \((0, 3)\), \((2, -1)\) and \((5, 3)\).

Step 2: Identify the diagonals

In a rhombus, the diagonals connect opposite vertices. So the diagonals here are between \((-3, -1)\) and \((5, 3)\), and between \((0, 3)\) and \((2, -1)\).

Step 3: Calculate the slope of the first diagonal

Use the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\) to find the slope of the diagonal connecting \((-3, -1)\) and \((5, 3)\): \[m_1 = \frac{3 - (-1)}{5 - (-3)} = \frac{4}{8} = \frac{1}{2}\]

Step 4: Calculate the slope of the second diagonal

Again, use the slope formula to find the slope of the diagonal connecting \((0, 3)\) and \((2, -1)\): \[m_2 = \frac{-1 - 3}{2 - 0} = \frac{-4}{2} = -2\]

Step 5: Conclude about the diagonals

Since the slopes of the diagonals are \(+\frac{1}{2}\) and \(-2\), their product is \(-1\) \((\frac{1}{2} \cdot -2 = -1)\). This means the diagonals are perpendicular to each other.

Key Concepts

quadrilaterals

A quadrilateral is a four-sided polygon. There are various types of quadrilaterals, with each type having unique properties.
The most common types include squares, rectangles, trapezoids, and rhombuses.

• A rhombus is a special kind of quadrilateral where all four sides have equal length.
• It also has opposite angles that are equal.
• Unlike a square, the corners of a rhombus are generally not 90 degrees.

Understanding these basic properties helps in solving geometry problems involving quadrilaterals, such as finding the slopes of diagonals.

slopes of lines

The slope of a line indicates its steepness and direction. It is calculated using the formula:
\(m = \frac{y_2 - y_1}{x_2 - x_1}\)

Here, \( (x_1, y_1)\) and \( (x_2, y_2)\) are coordinates of two points on the line.
In the context of a rhombus:

• Each diagonal connects opposite vertices.
• The slopes help us to understand the angle between these diagonals.

For example, if you have points \((-3, -1)\) and \(5, 3)\), you would plug these into the formula to find the slope:
\[ m_1 = \frac{3 - (-1)}{5 - (-3)} = \frac{4}{8} = \frac{1}{2} \] This shows a positive slope, indicating a line rising from left to right.

perpendicular diagonals

In a rhombus, the diagonals are always perpendicular to each other. This means they intersect at a 90-degree angle.

To confirm that two lines are perpendicular, you can check if the product of their slopes equals \(-1\).
For example:

• Diagonal 1 with a slope of \(\frac{1}{2}\).
• Diagonal 2 with a slope of \(-2\).

The product of these slopes is:
\(\frac{1}{2} \times -2 = -1\)
Thus, the diagonals are indeed perpendicular. This property is unique to rhombuses and squares, helping you identify these shapes easily in geometry problems.

coordinate geometry

Coordinate geometry, or analytic geometry, allows us to use algebraic equations to explore geometric concepts.
By placing shapes in a coordinate plane, you can calculate distances, slopes, and other properties using coordinates.

• For instance, identifying vertices of a rhombus \((x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4)\).
• Using the slope formula to find the slopes of the diagonals.
• Confirming properties such as sides being equal or diagonals being perpendicular.

This approach allows for precise calculations and clear visualizations of geometric shapes.