Answer :
To determine which statement proves that the parallelogram KLMN is a rhombus, we need to review the geometric properties of rhombuses. A parallelogram is specifically a rhombus if all its sides are equal in length and the diagonals bisect each other at right angles.
Here is a step-by-step explanation:
1. Statement a: "The midpoint of both diagonals is [tex]\((4,4)\)[/tex]."
- While this information confirms that the diagonals bisect each other and share a common midpoint, it does not prove that all four sides of the parallelogram are equal in length. Therefore, this statement alone does not confirm that KLMN is a rhombus.
2. Statement b: "The length of [tex]\(\overline{KM}\)[/tex] is [tex]\(\sqrt{72}\)[/tex] and the length of [tex]\(\overline{NL}\)[/tex] is [tex]\(\sqrt{8}\)[/tex]."
- This statement provides the lengths of the diagonals but does not indicate anything about the side lengths of the parallelogram. Hence, it does not help in proving that all sides are equal.
3. Statement c: "The slopes of [tex]\(\overline{LM}\)[/tex] and [tex]\(\overline{KN}\)[/tex] are both [tex]\(\frac{1}{2}\)[/tex] and [tex]\(NK = ML = \sqrt{20}\)[/tex]."
- This statement is crucial because it provides two pieces of information:
- The slopes of [tex]\(\overline{LM}\)[/tex] and [tex]\(\overline{KN}\)[/tex] are the same, which indicates that these segments are parallel and possibly part of a parallelogram.
- Most importantly, it mentions that [tex]\(NK\)[/tex] and [tex]\(ML\)[/tex] are both [tex]\(\sqrt{20}\)[/tex] in length, ensuring that at least two sides of the parallelogram are equal.
- In the context of a parallelogram, if we have two consecutive sides that are equal in length, it implies that all four sides must be equal (since opposite sides are also equal by definition of a parallelogram). Hence, this verifies that KLMN is a rhombus.
4. Statement d: "The slope of [tex]\(\overline{KM}\)[/tex] is 1 and the slope of [tex]\(\overline{NL}\)[/tex] is -1."
- This information tells us about the orientation of the diagonals; specifically, that they are perpendicular since the product of their slopes is [tex]\(-1\)[/tex]. While this property is true for rhombuses, it is also true for other shapes like rectangles and squares. This alone does not confirm the side lengths and therefore cannot prove the parallelogram is a rhombus.
Considering all the information, statement c is the most comprehensive in confirming that KLMN is a rhombus because it addresses both the parallel nature and the equal length of the sides sufficiently.
Therefore, the statement that proves that parallelogram KLMN is a rhombus is:
- c. The slopes of [tex]\(\overline{LM}\)[/tex] and [tex]\(\overline{KN}\)[/tex] are both [tex]\(\frac{1}{2}\)[/tex] and [tex]\(NK = ML = \sqrt{20}\)[/tex].
Thus, the correct answer is option 3, corresponding to statement c.
Here is a step-by-step explanation:
1. Statement a: "The midpoint of both diagonals is [tex]\((4,4)\)[/tex]."
- While this information confirms that the diagonals bisect each other and share a common midpoint, it does not prove that all four sides of the parallelogram are equal in length. Therefore, this statement alone does not confirm that KLMN is a rhombus.
2. Statement b: "The length of [tex]\(\overline{KM}\)[/tex] is [tex]\(\sqrt{72}\)[/tex] and the length of [tex]\(\overline{NL}\)[/tex] is [tex]\(\sqrt{8}\)[/tex]."
- This statement provides the lengths of the diagonals but does not indicate anything about the side lengths of the parallelogram. Hence, it does not help in proving that all sides are equal.
3. Statement c: "The slopes of [tex]\(\overline{LM}\)[/tex] and [tex]\(\overline{KN}\)[/tex] are both [tex]\(\frac{1}{2}\)[/tex] and [tex]\(NK = ML = \sqrt{20}\)[/tex]."
- This statement is crucial because it provides two pieces of information:
- The slopes of [tex]\(\overline{LM}\)[/tex] and [tex]\(\overline{KN}\)[/tex] are the same, which indicates that these segments are parallel and possibly part of a parallelogram.
- Most importantly, it mentions that [tex]\(NK\)[/tex] and [tex]\(ML\)[/tex] are both [tex]\(\sqrt{20}\)[/tex] in length, ensuring that at least two sides of the parallelogram are equal.
- In the context of a parallelogram, if we have two consecutive sides that are equal in length, it implies that all four sides must be equal (since opposite sides are also equal by definition of a parallelogram). Hence, this verifies that KLMN is a rhombus.
4. Statement d: "The slope of [tex]\(\overline{KM}\)[/tex] is 1 and the slope of [tex]\(\overline{NL}\)[/tex] is -1."
- This information tells us about the orientation of the diagonals; specifically, that they are perpendicular since the product of their slopes is [tex]\(-1\)[/tex]. While this property is true for rhombuses, it is also true for other shapes like rectangles and squares. This alone does not confirm the side lengths and therefore cannot prove the parallelogram is a rhombus.
Considering all the information, statement c is the most comprehensive in confirming that KLMN is a rhombus because it addresses both the parallel nature and the equal length of the sides sufficiently.
Therefore, the statement that proves that parallelogram KLMN is a rhombus is:
- c. The slopes of [tex]\(\overline{LM}\)[/tex] and [tex]\(\overline{KN}\)[/tex] are both [tex]\(\frac{1}{2}\)[/tex] and [tex]\(NK = ML = \sqrt{20}\)[/tex].
Thus, the correct answer is option 3, corresponding to statement c.
