In a parallelogram, opposite sides are of equal length. MANE is a prallelogram.
What is a parallelogram?
A parallelogram is a quadrilateral whose opposite sides are of equal length and are parallel to each other.
Given to us
M (6,20), A(10,8), N(4,0) and E (0,12).
We know that in a parallelogram, the pair of opposite sides are of equal length and parallel to each other. Therefore, the slope of the line ME and NA, and EN and MA should be equal.
Also, the length of the sides must be equal, therefore,
ME = NA
EN = MA
The slope of line ME and NA,
The formula to find the slope of a line is given by the formula,
[tex]m = \dfrac{y_2 - y_1}{x_2 - x_1}[/tex]
The slope of line MA(m₁),
M = (6,20) = [tex](x_2,\ y_2)[/tex]
E = (0,12) = [tex](x_1,\ y_1)[/tex]
[tex]m_1 = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{20-12}{6-0} = \dfrac{8}{6}[/tex]
The slope of line NA(m₂),
N = (4, 0) = [tex](x_2,\ y_2)[/tex]
A = (10,8) = [tex](x_1,\ y_1)[/tex]
[tex]m_2 = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{0-8}{4-10} = \dfrac{8}{6}[/tex]
As the slope of both the line is equal, therefore, the lines are parallel.
The slope of line EN and MA,
The formula to find the slope of a line is given by the formula,
[tex]m = \dfrac{y_2 - y_1}{x_2 - x_1}[/tex]
The slope of line EN(m₃),
E = (0,12) = [tex](x_2,\ y_2)[/tex]
N = (4,0) = [tex](x_1,\ y_1)[/tex]
[tex]m_3 = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{12-0}{0-4} = -\dfrac{12}{4}[/tex]
The slope of line MA(m₄),
M = (6,20) = [tex](x_2,\ y_2)[/tex]
A = (10,8) = [tex](x_1,\ y_1)[/tex]
[tex]m_4 = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{20-8}{6-10} = -\dfrac{12}{4}[/tex]
As the slope of both the line is equal, therefore, the lines are parallel.
Now, the length of the line must be equal, and we know to find the length between any two points,
[tex]D= \sqrt{(y_2-y_1)^2+(x_2-x_1)^2}\\\\[/tex]
The length of line ME and NA,
The length of line MA,
M = (6,20) = [tex](x_2,\ y_2)[/tex]
E = (0,12) = [tex](x_1,\ y_1)[/tex]
[tex]D= \sqrt{(y_2-y_1)^2+(x_2-x_1)^2}= 10[/tex]
The length of line NA,
N = (4, 0) = [tex](x_2,\ y_2)[/tex]
A = (10,8) = [tex](x_1,\ y_1)[/tex]
[tex]D= \sqrt{(y_2-y_1)^2+(x_2-x_1)^2}= 10[/tex]
Thus, the length of both lines is equal.
The length of line EN and NA,
The length of line EN,
E = (0,12) = [tex](x_2,\ y_2)[/tex]
N = (4,0) = [tex](x_1,\ y_1)[/tex]
[tex]D= \sqrt{(y_2-y_1)^2+(x_2-x_1)^2}= \sqrt{160}[/tex]
The slope of line MA,
M = (6,20) = [tex](x_2,\ y_2)[/tex]
A = (10,8) = [tex](x_1,\ y_1)[/tex]
[tex]D= \sqrt{(y_2-y_1)^2+(x_2-x_1)^2}= \sqrt{160}[/tex]
Thus, the length of both lines is equal.
Hence, MANE is a prallelogram.
Learn more about a Parallelogram:
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