To determine whether the given vertices \( N(-2,3) \), \( A(6,7) \), \( V(2,-1) \), and \( Y(-6,-5) \) form a rhombus, we first need to calculate the slopes of the diagonals \( NV \) and \( AY \). If the diagonals are perpendicular, it will confirm that the quadrilateral is a rhombus.
Step 1: Calculate the slope of diagonal \( NV \)
The slope \( m \) is given by the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For \( N(-2,3) \) and \( V(2,-1) \):
\[ m_{NV} = \frac{-1 - 3}{2 - (-2)} = \frac{-4}{4} = -1 \]
Step 2: Calculate the slope of diagonal \( AY \)
Using the same formula for points \( A(6,7) \) and \( Y(-6,-5) \):
\[ m_{AY} = \frac{-5 - 7}{-6 - 6} = \frac{-12}{-12} = 1 \]
Step 3: Check if the diagonals are perpendicular
Two lines are perpendicular if the product of their slopes equals -1.
Calculate the product of the slopes:
\[ m_{NV} \times m_{AY} = (-1) \times (1) = -1 \]
Conclusion
Since the product of the slopes is -1, the diagonals \( NV \) and \( AY \) are perpendicular.
Final Answer
- The slope of \( NV \) is \(-1\).
- The slope of \( AY \) is \(1\).
- Since the diagonals are perpendicular, this quadrilateral is a rhombus.
The answer to "is this a rhombus" is 1 (yes).
1 answer