Answer:
ABCD is a parallelogram.
Step-by-step explanation:
Given quadrilateral ABCD with vertices A(-3,1), B(-1,4), C(5,4) and D(3,1)
we have to give the most precise name for quadrilateral ABCD.
Using distance formula
[tex]Distance=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
[tex]AB=\sqrt{(-1+3)^2+(4-1)^2}=\sqrt{4+9}=\sqrt{13}units[/tex]
[tex]BC=\sqrt{(5+1)^2+(4-4)^2}=\sqrt{36}=\sqrt{6}units[/tex]
[tex]CD=\sqrt{(3-5)^2+(1-4)^2}=\sqrt{4+9}=\sqrt{13}units[/tex]
[tex]DA=\sqrt{(-3-3)^2+(1-1)^2}=\sqrt{36}=\sqrt{6}units[/tex]
Opposite sides of quadrilateral ABCD are equal.
Now, we find the slope of the sides of quadrilateral
[tex]\text{Slope of AB=}\frac{y_2-y_1}{x_2-x_1}=\frac{4-1}{-1+3}=\frac{3}{2}[/tex]
[tex]\text{Slope of BC=}\frac{y_2-y_1}{x_2-x_1}=\frac{4-4}{5+1}=0[/tex]
[tex]\text{Slope of CD=}\frac{y_2-y_1}{x_2-x_1}=\frac{1-4}{3-5}=\frac{3}{2}[/tex]
[tex]\text{Slope of DA=}\frac{y_2-y_1}{x_2-x_1}=\frac{1-1}{-3-3}=0[/tex]
Slopes of opposite sides are equal.
⇒ The sides of opposite sides of quadrilateral are parallel and equal.
⇒ ABCD is a parallelogram.
Option b) is correct.
