Respuesta :
Answer: The correct option is (5). ABCD is a rhombus with non-perpendicular adjacent sides.
Step-by-step explanation: Given that the vertices of a quadrilateral are A(4, 5), B(2, 4), C(4, 3), and D(6, 4).
We are to select the correct statement from the given options.
The lengths of the sides of the quadrilateral ABCD are calculated using the distance formula as follows:
[tex]AB=\sqrt{(4-2)^2+(5-4)^2}=\sqrt{4+1}=\sqrt{5}~\textup{units},\\\\BC=\sqrt{(2-4)^2+(4-3)^2}=\sqrt{4+1}=\sqrt{5}~\textup{units},\\\\CD=\sqrt{(4-6)^2+(3-4)^2}=\sqrt{4+1}=\sqrt{5}~\textup{units},\\\\DA=\sqrt{(6-4)^2+(4-5)^2}=\sqrt{4+1}=\sqrt{5}~\textup{units}.[/tex]
Since AB = BC = CD = DA, so all the sides are congruent.
Now, the slopes of the sides are calculated as follows:
[tex]\textup{Slope of AB, }m=\dfrac{4-5}{2-4}=\dfrac{1}{2},\\\\\textup{Slope of BC, }n=\dfrac{3-4}{4-2}=-\dfrac{1}{2},\\\\\textup{Slope of CD, }o=\dfrac{4-3}{6-4}=\dfrac{1}{2},\\\\\textup{Slope of DA, }p=\dfrac{4-5}{6-4}=-\dfrac{1}{2}.[/tex]
Since, m = o, n = p and m × n = n × o = o × p = p × m ≠ - 1, so
the opposite sides are parallel and the adjacent sides are non-perpendicular.
Therefore, the quadrilateral ABCD is a RHOMBUS with each side congruent and adjacent sides non-perpendicular.
Thus, (5) is the correct option.
