Which statement proves that parallelogram KLMN is a rhombus?
The midpoint of both diagonals is $(4,4)$.
The length of $\overline{ KM }$ is $\sqrt{72}$ and the length of $\overline{ NL }$ is $\sqrt{8}$.
The slopes of $\overline{ LM }$ and $\overline{ KN }$ are both $\frac{1}{2}$ and $NK = ML$ =$\sqrt{20}$.
The slope of $\overline{ KM }$ is 1 and the slope of $\overline{ NL }$ is -1.
Answer (2)
Statement 1 only proves KLMN is a parallelogram.
Statement 2 indicates the diagonals are unequal, not proving it's a rhombus.
Statement 3 confirms KLMN is a parallelogram but doesn't prove it's a rhombus.
Statement 4 shows the diagonals are perpendicular, proving KLMN is a rhombus: The slope of K M is 1 and the slope of N L is -1.
Explanation
Problem Analysis Let's analyze each statement to determine which one proves that parallelogram KLMN is a rhombus. A rhombus is a parallelogram with all four sides equal in length, or equivalently, a parallelogram whose diagonals are perpendicular.
Analyzing Statement 1 Statement 1: The midpoint of both diagonals is (4,4). This statement only proves that KLMN is a parallelogram because it confirms that the diagonals bisect each other. It doesn't provide any information about the side lengths or angles.
Analyzing Statement 2 Statement 2: The length of K M is 72 and the length of N L is 8 . This statement tells us that the diagonals are not equal in length. While this means the parallelogram is not a square, it doesn't tell us whether the parallelogram is a rhombus. A rhombus requires all sides to be equal, or the diagonals to be perpendicular, and this statement provides neither.
Analyzing Statement 3 Statement 3: The slopes of L M and K N are both 2 1 and N K = M L = 20 . This statement only confirms that L M and K N are parallel and equal in length, which is consistent with KLMN being a parallelogram. However, it doesn't provide any information about whether adjacent sides are equal or whether the diagonals are perpendicular.
Analyzing Statement 4 Statement 4: The slope of K M is 1 and the slope of N L is -1. If the product of the slopes of two lines is -1, then the lines are perpendicular. In this case, the diagonals K M and N L are perpendicular because 1 × − 1 = − 1 . A parallelogram with perpendicular diagonals is a rhombus. Therefore, this statement proves that parallelogram KLMN is a rhombus.
Final Answer Therefore, the statement that proves that parallelogram KLMN is a rhombus is: The slope of K M is 1 and the slope of N L is -1.
Examples
Understanding the properties of parallelograms and rhombuses is useful in various real-world applications, such as architecture and engineering. For example, when designing structures with specific angles and side lengths, knowing that a parallelogram with perpendicular diagonals is a rhombus helps ensure structural integrity and aesthetic appeal. Imagine designing a decorative tile pattern using rhombuses; ensuring the diagonals are perpendicular guarantees that the tiles fit together perfectly, creating a visually appealing and structurally sound design.
The statement that proves parallelogram KLMN is a rhombus is the one indicating the slopes of diagonals ar{KM} and ar{NL} being 1 and -1, respectively. This confirms the diagonals are perpendicular. Only perpendicular diagonals confirm that a parallelogram is a rhombus.
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