Answer:
Part 1) The graph in the attached figure
Part 2) The perimeter is [tex]P=20\ units[/tex]
Part 3) The figure is a square
Part 4) The area is [tex]A=20\ units^2[/tex]
Step-by-step explanation:
step 1
Plot the quadrilateral
we have
A(10,-5),B(6,-8),C(3,-4),D(7,-1)
using a graphing tool
The quadrilateral in the attached figure
see the attached figure
step 2
Find the perimeter of the figure
The perimeter of the figure is equal to
[tex]P=AB+BC+CD+AD[/tex]
the formula to calculate the distance between two points is equal to
[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]
Find the length side AB
we have
A(10,-5),B(6,-8)
substitute in the formula
[tex]d=\sqrt{(-8+5)^{2}+(6-10)^{2}}[/tex]
[tex]d=\sqrt{(-3)^{2}+(-4)^{2}}[/tex]
[tex]d_A_B=5\ units[/tex]
Find the length side BC
we have
B(6,-8),C(3,-4)
substitute in the formula
[tex]d=\sqrt{(-4+8)^{2}+(3-6)^{2}}[/tex]
[tex]d=\sqrt{(4)^{2}+(-3)^{2}}[/tex]
[tex]d_B_C=5\ units[/tex]
Find the length side CD
we have
C(3,-4),D(7,-1)
substitute in the formula
[tex]d=\sqrt{(-1+4)^{2}+(7-3)^{2}}[/tex]
[tex]d=\sqrt{(3)^{2}+(4)^{2}}[/tex]
[tex]d_C_D=5\ units[/tex]
Find the length side AD
we have
A(10,-5),D(7,-1)
substitute in the formula
[tex]d=\sqrt{(-1+5)^{2}+(7-10)^{2}}[/tex]
[tex]d=\sqrt{(4)^{2}+(-3)^{2}}[/tex]
[tex]d_A_D=5\ units[/tex]
so
The perimeter is equal to
[tex]P=5+5+5+5=20\ units[/tex]
step 3
Determine the slope of the sides
Remember that
If two lines are perpendicular, then their slopes are opposite reciprocal (the product is equal to -1)
If two lines are parallel, then their slopes are the same
The formula to calculate the slope between two points is equal to
[tex]m=\frac{y2-y1}{x2-x1}[/tex]
Find the slope AB
we have
A(10,-5),B(6,-8)
substitute
[tex]m_A_B=\frac{-8+5}{6-10}=3/4[/tex]
Find the slope BC
we have
B(6,-8),C(3,-4)
substitute
[tex]m_B_C=\frac{-4+8}{3-6}=-4/3[/tex]
Find the slope CD
we have
C(3,-4),D(7,-1)
substitute
[tex]m_C_D=\frac{-1+4}{7-3}=3/4[/tex]
Find the slope AD
we have
A(10,-5),D(7,-1)
substitute
[tex]m_A_D=\frac{-1+5}{7-10}=-4/3[/tex]
Compare the slopes
[tex]m_A_B=m_C_D[/tex]
[tex]m_B_C=m_A_D[/tex]
[tex]m_A_B*m_A_D=-1[/tex]
so
Opposite sides are parallel and consecutive sides are perpendicular
The length sides are congruent
therefore
The quadrilateral is a square
step 4
Find the area
The area of a square is
[tex]A=4b[/tex]
where
b is the length side of the square
we have
[tex]b=5\ units[/tex]
[tex]A=4(5)=20\ units^2[/tex]
