Prove that the following vertices form a parallelogram by showing that the opposite sides are parallel.

Question

J(−8,−2), U(2,8), N(8,2) and K(−2,−8)

(2 points)
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The slope of JU is Response area

The slope of UN is Response area

The slope of NK is Response area

The slope of KJ is Response area

Answer 1

To show that the vertices \( J(-8, -2) \), \( U(2, 8) \), \( N(8, 2) \), and \( K(-2, -8) \) form a parallelogram, we need to calculate the slopes of the opposite sides: \( JU \) and \( NK \), as well as \( UN \) and \( KJ \).

Step 1: Calculate the Slope of Each Side

1. Slope of JU: \[ \text{slope}_{JU} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - (-2)}{2 - (-8)} = \frac{8 + 2}{2 + 8} = \frac{10}{10} = 1 \]

2. Slope of UN: \[ \text{slope}_{UN} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 8}{8 - 2} = \frac{-6}{6} = -1 \]

3. Slope of NK: \[ \text{slope}_{NK} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-8 - 2}{-2 - 8} = \frac{-10}{-10} = 1 \]

4. Slope of KJ: \[ \text{slope}_{KJ} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - (-8)}{-8 - (-2)} = \frac{-2 + 8}{-8 + 2} = \frac{6}{-6} = -1 \]

Step 2: Summarize the Results

• The slope of \( JU \) is 1.
• The slope of \( UN \) is -1.
• The slope of \( NK \) is 1.
• The slope of \( KJ \) is -1.

Conclusion

The opposite sides \( JU \) and \( NK \) have the same slope, as well as \( UN \) and \( KJ \) having the same slope. Therefore, the opposite sides are parallel, which confirms that the quadrilateral formed by points \( J, U, N, \) and \( K \) is indeed a parallelogram.

Fill in the Responses

• The slope of JU is 1
• The slope of UN is -1
• The slope of NK is 1
• The slope of KJ is -1