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 provides information about the lengths of the diagonals but doesn't prove KLMN is a rhombus.
Statement 3 proves KLMN is a parallelogram but not necessarily a rhombus.
Statement 4 proves the diagonals are perpendicular, which means KLMN is a rhombus. Therefore, the answer is statement 4.
Explanation
Analyzing the problem Let's analyze each statement to determine which one proves that parallelogram KLMN is a rhombus. Remember, a rhombus is a parallelogram with all four sides equal in length, or equivalently, a parallelogram with perpendicular diagonals.
Analyzing Statement 1 Statement 1: The midpoint of both diagonals is (4,4). This statement only confirms that KLMN is a parallelogram because the diagonals bisect each other. It doesn't provide any information about the side lengths or angles, so it doesn't prove that KLMN is a rhombus.
Analyzing Statement 2 Statement 2: The length of K M is 72 and the length of N L is 8 . This tells us that the diagonals are not equal in length. While a rhombus isn't required to have equal diagonals, this information alone doesn't prove or disprove that KLMN is a rhombus.
Analyzing Statement 3 Statement 3: The slopes of L M and K N are both 2 1 and N K = M L = 20 . The equal slopes of L M and K N confirm that these sides are parallel. The equal lengths N K = M L = 20 confirm that opposite sides are equal. However, this only proves that KLMN is a parallelogram. It doesn't tell us anything about the adjacent sides or the angles, so it doesn't prove that KLMN is a rhombus.
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 the diagonals is -1, then the diagonals are perpendicular. Let m 1 be the slope of K M and m 2 be the slope of N L . Then, m 1 = 1 and m 2 = − 1 . Since m 1 ⋅ m 2 = 1 ⋅ ( − 1 ) = − 1 , the diagonals are perpendicular. A parallelogram with perpendicular diagonals is a rhombus. Therefore, this statement proves that KLMN is a rhombus.
Final Answer Statement 4 proves that parallelogram KLMN is a rhombus because it states that the diagonals are perpendicular.
Examples
Understanding the properties of parallelograms and rhombuses is useful in various fields, such as architecture and engineering. For example, when designing a building with a parallelogram-shaped facade, knowing the conditions that make it a rhombus ensures that the structure has specific symmetrical properties, which can be important for both aesthetic and structural reasons. In this case, ensuring the diagonals are perpendicular would guarantee the parallelogram is a rhombus, providing equal side lengths and balanced visual appeal.
Statement 4 proves that parallelogram KLMN is a rhombus as it shows the diagonals are perpendicular. Other statements either confirm it as a parallelogram or do not provide sufficient conditions. Thus, the chosen option is Statement 4.
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