Question

the points ( k(1,-3), l(7,-7), m(5,-2) ), and ( n(-1,1) ) form a quadrilateral. find the desired slopes and lengths, then fill in the words that best identifies the type of quadrilateral.
answer
slope of ( overline{kl} = ) length of ( overline{kl} =)
slope of ( overline{lm} = ) length of ( overline{lm} =)
slope of ( overline{mn} = ) length of ( overline{mn} =)
slope of ( overline{nk} = ) length of ( overline{nk} =)
quadrilateral klmn is

Explanation:

Step1: Calculate the slope of \(\overline{KL}\)

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For \(K(1,-3)\) and \(L(7,-7)\), \(m_{KL}=\frac{-7-(-3)}{7 - 1}=\frac{-7 + 3}{6}=\frac{-4}{6}=-\frac{2}{3}\)

Step2: Calculate the length of \(\overline{KL}\)

The distance formula is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For \(K(1,-3)\) and \(L(7,-7)\), \(d_{KL}=\sqrt{(7 - 1)^2+(-7+3)^2}=\sqrt{36 + 16}=\sqrt{52}=2\sqrt{13}\)

Step3: Calculate the slope of \(\overline{LM}\)

For \(L(7,-7)\) and \(M(5,-2)\), \(m_{LM}=\frac{-2+7}{5 - 7}=\frac{5}{-2}=-\frac{5}{2}\)

Step4: Calculate the length of \(\overline{LM}\)

For \(L(7,-7)\) and \(M(5,-2)\), \(d_{LM}=\sqrt{(5 - 7)^2+(-2 + 7)^2}=\sqrt{4+25}=\sqrt{29}\)

Step5: Calculate the slope of \(\overline{MN}\)

For \(M(5,-2)\) and \(N(-1,1)\), \(m_{MN}=\frac{1+2}{-1 - 5}=\frac{3}{-6}=-\frac{1}{2}\)

Step6: Calculate the length of \(\overline{MN}\)

For \(M(5,-2)\) and \(N(-1,1)\), \(d_{MN}=\sqrt{(-1 - 5)^2+(1 + 2)^2}=\sqrt{36+9}=\sqrt{45}=3\sqrt{5}\)

Step7: Calculate the slope of \(\overline{NK}\)

For \(N(-1,1)\) and \(K(1,-3)\), \(m_{NK}=\frac{-3 - 1}{1+1}=\frac{-4}{2}=-2\)

Step8: Calculate the length of \(\overline{NK}\)

For \(N(-1,1)\) and \(K(1,-3)\), \(d_{NK}=\sqrt{(1 + 1)^2+(-3 - 1)^2}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5}\)

Now, let's check the slopes for parallelism.
We know that for two lines to be parallel \(m_1=m_2\).
\(m_{KL}=-\frac{2}{3}\), \(m_{LM}=-\frac{5}{2}\), \(m_{MN}=-\frac{1}{2}\), \(m_{NK}=-2\)

Let's check the product of slopes for perpendicularity. For two lines to be perpendicular \(m_1\times m_2=- 1\)
\(m_{KL}\times m_{NK}=(-\frac{2}{3})\times(-2)=\frac{4}{3}
eq - 1\)
\(m_{LM}\times m_{MN}=(-\frac{5}{2})\times(-\frac{1}{2})=\frac{5}{4}
eq - 1\)

Since no two sides are parallel and no two sides are perpendicular, we can't classify it as a parallelogram, rectangle, rhombus, square or trapezoid.

Answer:

slope of \(\overline{KL}=-\frac{2}{3}\), length of \(\overline{KL}=2\sqrt{13}\)
slope of \(\overline{LM}=-\frac{5}{2}\), length of \(\overline{LM}=\sqrt{29}\)
slope of \(\overline{MN}=-\frac{1}{2}\), length of \(\overline{MN}=3\sqrt{5}\)
slope of \(\overline{NK}=-2\), length of \(\overline{NK}=2\sqrt{5}\)
Quadrilateral \(KLMN\) is a general quadrilateral.