Question

O determine whether the figure with vertices S(−2,1), R(1,3), T(2,0), and Z(−1,−2) is a parallelogram, please select the correct option and provide a justification:

A. The figure is a parallelogram because opposite sides are congruent.
B. The figure is a parallelogram because opposite sides have equal slopes.
C. The figure is not a parallelogram because opposite sides have different lengths.
D. The figure is not a parallelogram because opposite sides have different slopes.

Answer 1

To determine if the figure with vertices S(-2,1), R(1,3), T(2,0), and Z(-1,-2) is a parallelogram, we need to check for congruent opposite sides and equal slopes. After calculating the lengths of the sides and the slopes, it is observed that the figure satisfies these conditions, indicating that it is indeed a parallelogram.

Step-by-step explanation:

To determine whether the figure with vertices S(−2,1), R(1,3), T(2,0), and Z(−1,−2) is a parallelogram, we need to check if the opposite sides are congruent and if the opposite sides have equal slopes. Let's calculate the lengths of the sides and the slopes of the sides:

1. Length of SR: √[(1 - (-2))^2 + (3 - 1)^2] = 3.61
2. Length of RT: √[(2 - 1)^2 + (0 - 3)^2] = 3.16
3. Length of TZ: √[(-1 - 2)^2 + (-2 - 0)^2] = 3.61
4. Length of ZS: √[(-2 - (-1))^2 + (1 - (-2))^2] = 3.16
5. Slope of SR: (3 - 1) / (1 - (-2)) = 2/3
6. Slope of RT: (0 - 3) / (2 - 1) = -3
7. Slope of TZ: (-2 - 0) / (-1 - 2) = 2/3
8. Slope of ZS: (1 - (-2)) / (-2 - (-1)) = -3

From the calculations, we can see that:

• Opposite sides SR and TZ have the same length.
• Opposite sides RT and ZS have the same length.
• Opposite sides SR and TZ have the same slope.
• Opposite sides RT and ZS have the same slope.

Based on these observations, we can conclude that the figure is a parallelogram because opposite sides are congruent and opposite sides have equal slopes.