Question

2. The following shape is graphed with vertices at R(-1,2), S(3,1), T(5,-3), and U(1,-2). Prove if this shape has any pairs of parallel sides that are also congruent or if no sides are parallel and congruent.

Answer 1

Hello, this is Bing. I can help you with this geometry problem.

To prove if the shape has any pairs of parallel sides that are also congruent, we can use the following steps:

- Step 1: Calculate the slopes and lengths of each side using the coordinates of the vertices. We can use the slope formula and the distance formula to do this. The results are shown in the table below:

| Side | Slope | Length |

| --- | --- | --- |

| R S ¯ | - 1 4 | 4.12 |

| S T ¯ | - 1 | 5.66 |

| T U ¯ | - 1 4 | 4.12 |

| U R ¯ | - 1 | 5.66 |

- Step 2: Compare the slopes and lengths of the opposite sides to see if they are parallel and congruent. Two lines are parallel if they have the same slope¹. Two segments are congruent if they have the same length. The results are shown in the table below:

| Opposite Sides | Slopes | Parallel? | Lengths | Congruent? |

| --- | --- | --- | --- | --- |

| R S ¯ and T U ¯ | - 1 4 and - 1 4 | Yes | 4.12 and 4.12 | Yes |

| S T ¯ and U R ¯ | - 1 and - 1 | Yes | 5.66 and 5.66 | Yes |

- Step 3: Conclude that the shape has two pairs of parallel sides that are also congruent, based on the results from step 2. This means that the shape is a parallelogram, according to the parallel congruent sides theorem.

Therefore, we have proved that the shape has two pairs of parallel sides that are also congruent, and that it is a parallelogram.