Question

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 R(-5, 6), A(4, 3), M(7, -6), P(-2, -3) Item 19 (2 points) Item 20 Make sure to simiplify any fractions! Item 21 The slope of RM is Item 22 The slope of AP is Item 23 Is this a RHOMBUS (Type 1 for Yes and Type 2 for No)?

Answer 1

To determine whether the vertices R(-5, 6), A(4, 3), M(7, -6), and P(-2, -3) form a rhombus, we need to check the slopes of the diagonals RM and AP, and determine if they are perpendicular.

### Step 1: Calculate the slopes of diagonals RM and AP

**Slope formula:**
The slope \( m \) of a line through points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

**Finding the slope of diagonal RM:**
- Points R(-5, 6) and M(7, -6)
\[
m_{RM} = \frac{-6 - 6}{7 - (-5)} = \frac{-12}{12} = -1
\]

**Finding the slope of diagonal AP:**
- Points A(4, 3) and P(-2, -3)
\[
m_{AP} = \frac{-3 - 3}{-2 - 4} = \frac{-6}{-6} = 1
\]

### Step 2: Check if RM and AP are perpendicular

Two lines are perpendicular if the product of their slopes is -1:
\[
m_{RM} \times m_{AP} = -1 \quad \text{(RHOMBUS condition)}
\]

Substituting the slopes we found:
\[
(-1) \times (1) = -1
\]

Since the product of the slopes is -1, the diagonals RM and AP are indeed perpendicular.

### Conclusion:
Since the diagonals are perpendicular, we can conclude that the given vertices form a rhombus.

Therefore, for the question prompts:
- The slope of RM is **-1**.
- The slope of AP is **1**.
- **Is this a RHOMBUS?** (Type 1 for Yes and Type 2 for No): **1** (Yes).