Question

question
the points ( n(7,-2), o(2,-4) ), and ( p(-2,6) ) form a triangle. plot the points then click the \graph triangle\ button.
click on the graph to plot a point. click a point to delete it.
answer attempt 1 out of a
find the desired slopes and lengths, then fill in the words that characterize the triangle.
slope of ( overline{no}= ) length of ( overline{no}= )
slope of ( overline{op}= ) length of ( overline{op}= )
slope of ( overline{pn}= ) length of ( overline{pn}= )

Explanation:

Step1: Calculate the slope of \(\overline{NO}\)

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
For points \(N(7,-2)\) and \(O(2,-4)\), \(x_1 = 7,y_1=-2,x_2 = 2,y_2=-4\).
\(m_{NO}=\frac{-4-(-2)}{2 - 7}=\frac{-4 + 2}{-5}=\frac{-2}{-5}=\frac{2}{5}\).

Step2: Calculate the length of \(\overline{NO}\)

The distance formula is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
\(d_{NO}=\sqrt{(2 - 7)^2+(-4+2)^2}=\sqrt{(-5)^2+(-2)^2}=\sqrt{25 + 4}=\sqrt{29}\).

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

For points \(O(2,-4)\) and \(P(-2,6)\), \(x_1 = 2,y_1=-4,x_2=-2,y_2 = 6\).
\(m_{OP}=\frac{6-(-4)}{-2 - 2}=\frac{6 + 4}{-4}=\frac{10}{-4}=-\frac{5}{2}\).

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

\(d_{OP}=\sqrt{(-2 - 2)^2+(6 + 4)^2}=\sqrt{(-4)^2+10^2}=\sqrt{16+100}=\sqrt{116}=2\sqrt{29}\).

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

For points \(P(-2,6)\) and \(N(7,-2)\), \(x_1=-2,y_1 = 6,x_2 = 7,y_2=-2\).
\(m_{PN}=\frac{-2-6}{7+2}=\frac{-8}{9}\).

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

\(d_{PN}=\sqrt{(7 + 2)^2+(-2 - 6)^2}=\sqrt{9^2+(-8)^2}=\sqrt{81+64}=\sqrt{145}\).

Answer:

slope of \(\overline{NO}=\frac{2}{5}\), length of \(\overline{NO}=\sqrt{29}\), slope of \(\overline{OP}=-\frac{5}{2}\), length of \(\overline{OP}=2\sqrt{29}\), slope of \(\overline{PN}=-\frac{8}{9}\), length of \(\overline{PN}=\sqrt{145}\)