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.

N(-2,3), A(6,7) V(2,-1), Y(-6,-5)



Make sure to simplify any fractions

the slope of NV is_____

the slope of AY is______
is this a rhombus (type 1 for yes and 2 for no)?

Answer 1

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)**.