Question

a certain polygon has its vertices at the following points: (2, 5), (4, 7), (7, 7), and (9, 5)
what is the best description of this polygon?
a. pentagon
b. trapezoid
c. rhombus
d. parallelogram

Explanation:

Step1: Recall polygon - vertex count

A pentagon has 5 vertices. Given polygon has 4 vertices, so not a pentagon.

Step2: Calculate slopes of sides

Let the points be \(A(2,5)\), \(B(4,7)\), \(C(7,7)\), \(D(9,5)\).
The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
Slope of \(AB\): \(m_{AB}=\frac{7 - 5}{4 - 2}=\frac{2}{2} = 1\).
Slope of \(BC\): \(m_{BC}=\frac{7 - 7}{7 - 4}=0\).
Slope of \(CD\): \(m_{CD}=\frac{5 - 7}{9 - 7}=\frac{-2}{2}=-1\).
Slope of \(DA\): \(m_{DA}=\frac{5 - 5}{2 - 9}=0\).

Step3: Analyze parallel - sides

Since \(m_{BC}=m_{DA} = 0\), \(BC\) is parallel to \(DA\). And the other two non - parallel sides \(AB\) and \(CD\) have non - equal non - zero slopes. A trapezoid has one pair of parallel sides. So the polygon is a trapezoid.
A rhombus has all sides equal and opposite sides parallel. A parallelogram has two pairs of parallel sides. This polygon does not meet those criteria.

Answer:

B. trapezoid