To prove that the vertices \( J(-8, -2) \), \( U(2, 8) \), \( N(8, 2) \), and \( K(-2, -8) \) form a parallelogram, we need to show that the opposite sides of the quadrilateral formed by these points are parallel. We can do this by calculating the slopes of the sides and verifying that the slopes of opposite sides are equal.
Step 1: Calculate the slopes of the sides
The slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
1. Slope of side JU:
- Points \( J(-8, -2) \) and \( U(2, 8) \): \[ m_{JU} = \frac{8 - (-2)}{2 - (-8)} = \frac{8 + 2}{2 + 8} = \frac{10}{10} = 1 \]
2. Slope of side NK:
- Points \( N(8, 2) \) and \( K(-2, -8) \): \[ m_{NK} = \frac{-8 - 2}{-2 - 8} = \frac{-10}{-10} = 1 \]
3. Slope of side U N:
- Points \( U(2, 8) \) and \( N(8, 2) \): \[ m_{UN} = \frac{2 - 8}{8 - 2} = \frac{-6}{6} = -1 \]
4. Slope of side JK:
- Points \( J(-8, -2) \) and \( K(-2, -8) \): \[ m_{JK} = \frac{-8 - (-2)}{-2 - (-8)} = \frac{-8 + 2}{-2 + 8} = \frac{-6}{6} = -1 \]
Step 2: Compare the slopes
Now we can compare the slopes of the opposite sides:
- Slope of \( JU \) is \( 1 \) and slope of \( NK \) is \( 1 \). Since \( m_{JU} = m_{NK} \), these sides are parallel.
- Slope of \( UN \) is \( -1 \) and slope of \( JK \) is \( -1 \). Since \( m_{UN} = m_{JK} \), these sides are also parallel.
Conclusion
Since both pairs of opposite sides are parallel, we conclude that the quadrilateral formed by the points \( J \), \( U \), \( N \), and \( K \) is a parallelogram.
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