Question

find the desired slopes and lengths, then fill in the words that characterize the triangle.
answer attempt 1 out of 2
slope of \\( \overline{cd}=0 \\) slope of \\( \overline{de}=\\) slope of \\( \overline{ec}=\\)
length of \\( \overline{cd}=\\) length of \\( \overline{de}=\\) length of \\( \overline{ec}=\\)
triangle cde is

Explanation:

Step1: Find coordinates of points

From the graph, assume \(C(2, - 1)\), \(D(-4,-1)\), \(E(-1,-7)\)

Step2: Calculate slope of \(\overline{DE}\)

Slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For \(D(-4,-1)\) and \(E(-1,-7)\), \(m_{DE}=\frac{-7-( - 1)}{-1-( - 4)}=\frac{-7 + 1}{-1 + 4}=\frac{-6}{3}=-2\)

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

For \(E(-1,-7)\) and \(C(2,-1)\), \(m_{EC}=\frac{-1-( - 7)}{2-( - 1)}=\frac{-1 + 7}{2 + 1}=\frac{6}{3}=2\)

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

Distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For \(C(2,-1)\) and \(D(-4,-1)\), \(d_{CD}=\sqrt{(2-( - 4))^2+(-1-( - 1))^2}=\sqrt{(6)^2+(0)^2}=6\)

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

For \(D(-4,-1)\) and \(E(-1,-7)\), \(d_{DE}=\sqrt{(-1-( - 4))^2+(-7-( - 1))^2}=\sqrt{(3)^2+(-6)^2}=\sqrt{9 + 36}=\sqrt{45}=3\sqrt{5}\)

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

For \(E(-1,-7)\) and \(C(2,-1)\), \(d_{EC}=\sqrt{(2-( - 1))^2+(-1-( - 7))^2}=\sqrt{(3)^2+(6)^2}=\sqrt{9+36}=\sqrt{45}=3\sqrt{5}\)

Since \(DE = EC\), triangle \(CDE\) is isosceles.

Answer:

slope of \(\overline{DE}=-2\), slope of \(\overline{EC}=2\), length of \(\overline{CD}=6\), length of \(\overline{DE}=3\sqrt{5}\), length of \(\overline{EC}=3\sqrt{5}\), Triangle \(CDE\) is isosceles.