Question

the points ( g(6,-9), h(2,-2), i(-6,-1) ), and ( j(-2,-8) ) 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{gh} = ) length of ( overline{gh} =
slope of ( overline{hi} = ) length of ( overline{hi} =
slope of ( overline{ij} = ) length of ( overline{ij} =
slope of ( overline{jg} = ) length of ( overline{jg} =
quadrilateral ( ghij ) is

Explanation:

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

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For \(G(6,-9)\) and \(H(2,-2)\), \(m_{GH}=\frac{-2-(-9)}{2 - 6}=\frac{-2 + 9}{-4}=\frac{7}{-4}=-\frac{7}{4}\)

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

The distance formula is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For \(G(6,-9)\) and \(H(2,-2)\), \(d_{GH}=\sqrt{(2 - 6)^2+(-2+9)^2}=\sqrt{(-4)^2+7^2}=\sqrt{16 + 49}=\sqrt{65}\)

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

For \(H(2,-2)\) and \(I(-6,-1)\), \(m_{HI}=\frac{-1-(-2)}{-6 - 2}=\frac{-1 + 2}{-8}=\frac{1}{-8}=-\frac{1}{8}\)

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

For \(H(2,-2)\) and \(I(-6,-1)\), \(d_{HI}=\sqrt{(-6 - 2)^2+(-1 + 2)^2}=\sqrt{(-8)^2+1^2}=\sqrt{64+1}=\sqrt{65}\)

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

For \(I(-6,-1)\) and \(J(-2,-8)\), \(m_{IJ}=\frac{-8-(-1)}{-2+6}=\frac{-8 + 1}{4}=\frac{-7}{4}=-\frac{7}{4}\)

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

For \(I(-6,-1)\) and \(J(-2,-8)\), \(d_{IJ}=\sqrt{(-2 + 6)^2+(-8 + 1)^2}=\sqrt{4^2+(-7)^2}=\sqrt{16+49}=\sqrt{65}\)

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

For \(J(-2,-8)\) and \(G(6,-9)\), \(m_{JG}=\frac{-9-(-8)}{6+2}=\frac{-9 + 8}{8}=\frac{-1}{8}=-\frac{1}{8}\)

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

For \(J(-2,-8)\) and \(G(6,-9)\), \(d_{JG}=\sqrt{(6 + 2)^2+(-9 + 8)^2}=\sqrt{8^2+(-1)^2}=\sqrt{64 + 1}=\sqrt{65}\)

Since \(m_{GH}=m_{IJ}=-\frac{7}{4}\), \(m_{HI}=m_{JG}=-\frac{1}{8}\) and \(d_{GH}=d_{HI}=d_{IJ}=d_{JG}=\sqrt{65}\)

Answer:

slope of \(\overline{GH}=-\frac{7}{4}\), length of \(\overline{GH}=\sqrt{65}\)
slope of \(\overline{HI}=-\frac{1}{8}\), length of \(\overline{HI}=\sqrt{65}\)
slope of \(\overline{IJ}=-\frac{7}{4}\), length of \(\overline{IJ}=\sqrt{65}\)
slope of \(\overline{JG}=-\frac{1}{8}\), length of \(\overline{JG}=\sqrt{65}\)
Quadrilateral \(GHIJ\) is a rhombus