Question

the points ( e(2,-7) ), ( f(7,-4) ), ( g(4,1) ), and ( h(-1,-2) ) form a quadrilateral. find the desired slopes and lengths, then fill in the words that best identifies the type of quadrilateral.
answer attempt 1 out of 3
slope of ( overline{ef} ) =
length of ( overline{ef} ) =
slope of ( overline{fg} ) =
length of ( overline{fg} ) =
slope of ( overline{gh} ) =
length of ( overline{gh} ) =
slope of ( overline{he} ) =
length of ( overline{he} ) =
quadrilateral ( efgh ) is

Explanation:

Step1: Calculate slope of \(\overline{EF}\)

Slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For \(E(2,-7)\) and \(F(7,-4)\), \(m_{EF}=\frac{-4-(-7)}{7 - 2}=\frac{-4 + 7}{5}=\frac{3}{5}\)

Step2: Calculate length of \(\overline{EF}\)

Distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). \(d_{EF}=\sqrt{(7 - 2)^2+(-4+7)^2}=\sqrt{25 + 9}=\sqrt{34}\)

Step3: Calculate slope of \(\overline{FG}\)

For \(F(7,-4)\) and \(G(4,1)\), \(m_{FG}=\frac{1-(-4)}{4 - 7}=\frac{1 + 4}{-3}=-\frac{5}{3}\)

Step4: Calculate length of \(\overline{FG}\)

\(d_{FG}=\sqrt{(4 - 7)^2+(1 + 4)^2}=\sqrt{9+25}=\sqrt{34}\)

Step5: Calculate slope of \(\overline{GH}\)

For \(G(4,1)\) and \(H(-1,-2)\), \(m_{GH}=\frac{-2 - 1}{-1 - 4}=\frac{-3}{-5}=\frac{3}{5}\)

Step6: Calculate length of \(\overline{GH}\)

\(d_{GH}=\sqrt{(-1 - 4)^2+(-2 - 1)^2}=\sqrt{25 + 9}=\sqrt{34}\)

Step7: Calculate slope of \(\overline{HE}\)

For \(H(-1,-2)\) and \(E(2,-7)\), \(m_{HE}=\frac{-7-(-2)}{2+1}=\frac{-7 + 2}{3}=-\frac{5}{3}\)

Step8: Calculate length of \(\overline{HE}\)

\(d_{HE}=\sqrt{(2 + 1)^2+(-7 + 2)^2}=\sqrt{9+25}=\sqrt{34}\)

Since \(m_{EF}=m_{GH}=\frac{3}{5}\), \(m_{FG}=m_{HE}=-\frac{5}{3}\) (opposite sides are parallel) and \(d_{EF}=d_{FG}=d_{GH}=d_{HE}=\sqrt{34}\) (all sides are equal)

Answer:

slope of \(\overline{EF}=\frac{3}{5}\), length of \(\overline{EF}=\sqrt{34}\)
slope of \(\overline{FG}=-\frac{5}{3}\), length of \(\overline{FG}=\sqrt{34}\)
slope of \(\overline{GH}=\frac{3}{5}\), length of \(\overline{GH}=\sqrt{34}\)
slope of \(\overline{HE}=-\frac{5}{3}\), length of \(\overline{HE}=\sqrt{34}\)
Quadrilateral \(EFGH\) is a rhombus