Answer:
The most precise name for a quadrilateral ABCD is a parallelogram
Step-by-step explanation:
we have
A(2,3) B(7,2) C(6,-1) D(1,0)
Plot the quadrilateral'
using a graphing tool
The quadrilateral ABCD in the attached figure
Verify the length of the sides
the formula to calculate the distance between two points is equal to
[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]
step 1
Find distance AB
A(2,3) B(7,2)
substitute
[tex]d=\sqrt{(2-3)^{2}+(7-2)^{2}}[/tex]
[tex]d=\sqrt{(-1)^{2}+(5)^{2}}[/tex]
[tex]d_A_B=\sqrt{26}\ units[/tex]
step 2
Find distance BC
B(7,2) C(6,-1)
substitute
[tex]d=\sqrt{(-1-2)^{2}+(6-7)^{2}}[/tex]
[tex]d=\sqrt{(-3)^{2}+(-1)^{2}}[/tex]
[tex]d_B_C=\sqrt{10}\ units[/tex]
step 3
Find distance CD
C(6,-1) D(1,0)
substitute
[tex]d=\sqrt{(0+1)^{2}+(1-6)^{2}}[/tex]
[tex]d=\sqrt{(1)^{2}+(-5)^{2}}[/tex]
[tex]d_C_D=\sqrt{26}\ units[/tex]
step 4
Find distance AD
A(2,3) D(1,0)
substitute
[tex]d=\sqrt{(0-3)^{2}+(1-2)^{2}}[/tex]
[tex]d=\sqrt{(-3)^{2}+(-1)^{2}}[/tex]
[tex]d_A_D=\sqrt{10}\ units[/tex]
step 5
Compare the length sides
AB=CD
BC=AD
Opposite sides are congruent
Verify the slope of the sides
The formula to calculate the slope between two points is equal to
[tex]m=\frac{y2-y1}{x2-x1}[/tex]
step 1
Find slope AB
A(2,3) B(7,2)
substitute
[tex]m=\frac{2-3}{7-2}[/tex]
[tex]m=\frac{-1}{5}[/tex]
[tex]m_A_B=-\frac{1}{5}[/tex]
step 2
Find slope BC
B(7,2) C(6,-1)
substitute
[tex]m=\frac{-1-2}{6-7}[/tex]
[tex]m=\frac{-3}{-1}[/tex]
[tex]m_B_C=3[/tex]
step 3
Find slope CD
C(6,-1) D(1,0)
substitute
[tex]m=\frac{0+1}{1-6}[/tex]
[tex]m=\frac{1}{-5}[/tex]
[tex]m_C_D=-\frac{1}{5}[/tex]
step 4
Find slope AD
A(2,3) D(1,0)
substitute
[tex]m=\frac{0-3}{1-2}[/tex]
[tex]m=\frac{-3}{-1}[/tex]
[tex]m_A_D=3[/tex]
step 5
Compare the slopes
[tex]m_A_B=m_C_D[/tex]
[tex]m_B_C=m_A_D[/tex]
The slope of the opposite sides are equal, that means, opposite sides are parallel
The slopes of consecutive sides are not opposite reciprocal, that means, consecutive sides are not perpendicular
therefore
The most precise name for a quadrilateral ABCD is a parallelogram
