- A rhombus is a parallelogram with perpendicular diagonals, so verifying that the diagonals are perpendicular is sufficient to prove the parallelogram is a rhombus.
- The diagonal lengths and their shared midpoint are properties common to all parallelograms and do not specifically identify a rhombus.
- The product of the slopes of diagonals \(KM\) and \(NL\) is \(1 \times (-1) = -1\), confirming that the diagonals are perpendicular.
- The correct statement is: \(\boxed{\text{The slope of } \overline{KM} \text{ is } 1 \text{ and the slope of } \overline{NL} \text{ is } -1.}\)
Explanation
- Define properties of a parallelogram and a rhombus.
A parallelogram is a quadrilateral with two pairs of parallel sides. A rhombus is a special type of parallelogram where all four sides are equal in length. One of the primary geometric properties that distinguishes a rhombus from other parallelograms is the orientation of its diagonals: a parallelogram is a rhombus if and only if its diagonals are perpendicular to each other. We are given a parallelogram \(KLMN\) and asked which of the three provided statements proves that it is a rhombus.
- Evaluate the first statement.
The first statement provides the lengths of the diagonals: \(|KM| = \sqrt{72}\) and \(|NL| = \sqrt{8}\). In any parallelogram, the relationship between the lengths of the diagonals (\(d_1, d_2\)) and the side lengths (\(a, b\)) is given by the parallelogram law: \(d_1^2 + d_2^2 = 2(a^2 + b^2)\). Knowing the lengths of the diagonals alone does not force \(a = b\), which is required for the figure to be a rhombus. Therefore, this statement is insufficient to prove that the parallelogram is a rhombus.
- Evaluate the second statement.
The second statement states that the midpoint of both diagonals is \((4, 4)\). By the definition of a parallelogram, its diagonals must bisect each other, meaning they necessarily share the same midpoint. This property is true for every parallelogram, whether it is a rhombus, a rectangle, or a general parallelogram. Thus, this statement provides no unique information that would identify the figure as a rhombus.
- Evaluate the third statement and conclude.
The third statement provides the slopes of the diagonals: the slope of \(KM\) is \(1\) and the slope of \(NL\) is \(-1\). Two lines are perpendicular if and only if the product of their slopes is \(-1\). Calculating the product of the given slopes: \(1 \times (-1) = -1\). Since the diagonals are perpendicular, the parallelogram must satisfy the condition for being a rhombus. Therefore, this statement confirms that \(KLMN\) is a rhombus.
Examples
In architecture, when designing a support truss system, engineers often need to ensure that a four-sided frame is a rhombus to distribute loads evenly. By checking that the beams forming the diagonals are perfectly perpendicular (i.e., their slopes satisfy the condition \(m_1 \cdot m_2 = -1\)), they can verify the geometric integrity of the design, ensuring the structure acts as a stable rhombus rather than a general parallelogram.