Question

The following vertices form a parallelogram. Do the following vertices form a RHOMBUS also? Prove using the theorem that says IF the diagonals are PERPENDICULAR, then it is a rhombus. Item 17 Item 18 N(-2, 3), A(6, 7), (2, -1), Y(-6, -5) Item 19 (2 points) Item 20 Make sure to simiplify any fractions! Item 21 The slope of NV is Item 22 Item 23 The slope of AY is Item 24 < < Is this a RHOMBUS (Type 1 for Yes and Type 2 for No)?

Answer 1

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.

1. **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):
\[
m_{NV} = \frac{-1 - 3}{2 - (-2)} = \frac{-4}{4} = -1
\]

2. **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)