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 confirms KLMN is a parallelogram but doesn't prove it's a rhombus.
Statement 4 states the diagonals are perpendicular, which proves KLMN is a rhombus. Statement 4
Explanation
Analyze the properties of a rhombus. Let's analyze the given statements to determine which one proves that parallelogram KLMN is a rhombus. Recall that a rhombus is a parallelogram with all four sides equal in length, or equivalently, a parallelogram with perpendicular diagonals.
Analyze statement 1. Statement 1: The midpoint of both diagonals is (4,4). This statement only proves that the quadrilateral is a parallelogram, since 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.
Analyze 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 a rhombus isn't required to have equal diagonals (only a square does), this information alone doesn't prove or disprove that KLMN is a rhombus.
Analyze statement 3. Statement 3: The slopes of L M and K N are both 2 1 and N K = M L = 20 . Since the slopes of L M and K N are equal, L M ∥ K N . Also, N K = M L . This information confirms that KLMN is a parallelogram. However, it doesn't prove that adjacent sides are equal or that the diagonals are perpendicular, so it doesn't prove that KLMN is a rhombus.
Analyze statement 4. Statement 4: The slope of K M is 1 and the slope of N L is -1. If the diagonals have slopes that are negative reciprocals of each other, then the diagonals are perpendicular. Specifically, since 1 × − 1 = − 1 , the diagonals K M and N L are perpendicular. A parallelogram with perpendicular diagonals is a rhombus. Therefore, this statement proves that parallelogram KLMN is a rhombus.
Conclusion. Therefore, statement 4 proves that parallelogram KLMN is a rhombus.
Examples
Rhombuses are commonly found in architecture and design. For example, the Argyle pattern, often seen on sweaters and socks, is made up of overlapping rhombuses. Knowing the properties of a rhombus, such as having perpendicular diagonals, is useful in ensuring that these patterns are geometrically accurate and visually appealing. Also, in engineering, rhombus shapes can be used in structures where strength and stability are needed, such as in bridge designs.
Statement 4 proves that parallelogram KLMN is a rhombus by showing that the slopes of its diagonals are negative reciprocals, indicating they are perpendicular. The other statements either confirm that KLMN is a parallelogram or provide insufficient evidence regarding its classification as a rhombus. Therefore, the selected option is Statement 4.
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