Question

the points ( b(-5,0) ), ( c(-1,-3) ), ( d(2,1) ), and ( e(-2,4) ) 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{bc} = ) length of ( overline{bc} =
slope of ( overline{cd} = ) length of ( overline{cd} =
slope of ( overline{de} = ) length of ( overline{de} =
slope of ( overline{eb} = ) length of ( overline{eb} =
quadrilateral bcde is

Explanation:

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

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For points \(B(-5,0)\) and \(C(-1,-3)\), \(m_{BC}=\frac{-3 - 0}{-1-(-5)}=\frac{-3}{4}\)

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

The distance formula is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For \(B(-5,0)\) and \(C(-1,-3)\), \(d_{BC}=\sqrt{(-1 + 5)^2+(-3 - 0)^2}=\sqrt{16 + 9}=5\)

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

For points \(C(-1,-3)\) and \(D(2,1)\), \(m_{CD}=\frac{1-(-3)}{2-(-1)}=\frac{4}{3}\)

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

For \(C(-1,-3)\) and \(D(2,1)\), \(d_{CD}=\sqrt{(2 + 1)^2+(1 + 3)^2}=\sqrt{9+16}=5\)

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

For points \(D(2,1)\) and \(E(-2,4)\), \(m_{DE}=\frac{4 - 1}{-2-2}=\frac{3}{-4}=-\frac{3}{4}\)

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

For \(D(2,1)\) and \(E(-2,4)\), \(d_{DE}=\sqrt{(-2 - 2)^2+(4 - 1)^2}=\sqrt{16 + 9}=5\)

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

For points \(E(-2,4)\) and \(B(-5,0)\), \(m_{EB}=\frac{0 - 4}{-5-(-2)}=\frac{-4}{-3}=\frac{4}{3}\)

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

For \(E(-2,4)\) and \(B(-5,0)\), \(d_{EB}=\sqrt{(-5 + 2)^2+(0 - 4)^2}=\sqrt{9 + 16}=5\)

Since \(m_{BC}=m_{DE}=-\frac{3}{4}\), \(m_{CD}=m_{EB}=\frac{4}{3}\), \(d_{BC}=d_{CD}=d_{DE}=d_{EB} = 5\)

Answer:

slope of \(\overline{BC}=-\frac{3}{4}\), length of \(\overline{BC}=5\), slope of \(\overline{CD}=\frac{4}{3}\), length of \(\overline{CD}=5\), slope of \(\overline{DE}=-\frac{3}{4}\), length of \(\overline{DE}=5\), slope of \(\overline{EB}=\frac{4}{3}\), length of \(\overline{EB}=5\), Quadrilateral \(BCDE\) is a rhombus