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).