Question

a certain polygon has its vertices at the following points: (1, 1), (2, 5), (8, 1), and (9, 5). what is the best description of this polygon? a. square b. rhombus c. trapezoid d. parallelogram

Explanation:

Step1: Calculate slopes of sides

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$.
For side connecting $(1,1)$ and $(2,5)$: $m_1=\frac{5 - 1}{2 - 1}=4$.
For side connecting $(2,5)$ and $(9,5)$: $m_2=\frac{5 - 5}{9 - 2}=0$.
For side connecting $(9,5)$ and $(8,1)$: $m_3=\frac{1 - 5}{8 - 9}=4$.
For side connecting $(8,1)$ and $(1,1)$: $m_4=\frac{1 - 1}{1 - 8}=0$.

Step2: Analyze parallel - side condition

Since $m_1 = m_3$ and $m_2 = m_4$, opposite sides are parallel.
A square has all sides equal and all angles 90 degrees. Here, we haven't checked side - lengths and angles.
A rhombus has all sides equal, which we haven't checked.
A trapezoid has only one pair of parallel sides.
A parallelogram has two pairs of parallel sides, which is the case here. So the polygon is a parallelogram.

Answer:

D. parallelogram