To determine if the given vertices form a rhombus, we will first find the slopes of the diagonals formed by the vertices N(-2, 3), A(6, 7), Y(-6, -5), and V(2, -1).
Let's denote the vertices as follows:
- N = (-2, 3)
- A = (6, 7)
- Y = (-6, -5)
- V = (2, -1)
Step 1: Calculate the slopes of the diagonals.
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Slope of NV: The formula for the slope (m) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] For points N(-2, 3) and V(2, -1): \[ m_{NV} = \frac{-1 - 3}{2 - (-2)} = \frac{-4}{4} = -1 \]
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Slope of AY: For points A(6, 7) and Y(-6, -5): \[ m_{AY} = \frac{-5 - 7}{-6 - 6} = \frac{-12}{-12} = 1 \]
Step 2: Check if the diagonals are perpendicular.
Two lines are perpendicular if the product of their slopes is -1: \[ m_{NV} \times m_{AY} = -1 \times 1 = -1 \] Since the product is indeed -1, we conclude that the diagonals NV and AY are perpendicular.
Conclusion
According to the theorem mentioned, if the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.
Thus, we can conclude that the given vertices do form a rhombus.
Final Answer
Is this a RHOMBUS? Type 1 for Yes and Type 2 for No.
1 (Yes)