Question

the points l(2, -2), m(9, 0), and n(4, 3) form a triangle. plot the points then click the graph triangle
click on the graph to plot a point. click a point to delete it.
find the desired slopes and lengths, then fill in the words that characterize the triangle.
answer attempt 1 out of 2
slope of ( overline{lm} = ) slope of ( overline{mn} = ) slope of ( overline{nl} =
length of ( overline{lm} = ) length of ( overline{mn} = ) length of ( overline{nl} = )

Explanation:

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

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For points \(L(2,-2)\) and \(M(9,0)\), \(m_{LM}=\frac{0 - (-2)}{9 - 2}=\frac{2}{7}\)

Step2: Calculate the slope of \(\overline{MN}\)

For points \(M(9,0)\) and \(N(4,3)\), \(m_{MN}=\frac{3 - 0}{4 - 9}=\frac{3}{-5}=-\frac{3}{5}\)

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

For points \(N(4,3)\) and \(L(2,-2)\), \(m_{NL}=\frac{-2 - 3}{2 - 4}=\frac{-5}{-2}=\frac{5}{2}\)

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

The distance formula is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For \(L(2,-2)\) and \(M(9,0)\), \(d_{LM}=\sqrt{(9 - 2)^2+(0 + 2)^2}=\sqrt{49 + 4}=\sqrt{53}\)

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

For \(M(9,0)\) and \(N(4,3)\), \(d_{MN}=\sqrt{(4 - 9)^2+(3 - 0)^2}=\sqrt{25 + 9}=\sqrt{34}\)

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

For \(N(4,3)\) and \(L(2,-2)\), \(d_{NL}=\sqrt{(2 - 4)^2+(-2 - 3)^2}=\sqrt{4 + 25}=\sqrt{29}\)

Answer:

slope of \(\overline{LM}=\frac{2}{7}\), slope of \(\overline{MN}=-\frac{3}{5}\), slope of \(\overline{NL}=\frac{5}{2}\), length of \(\overline{LM}=\sqrt{53}\), length of \(\overline{MN}=\sqrt{34}\), length of \(\overline{NL}=\sqrt{29}\)