Question

the points ( n(0,-3), o(3,4) ), and ( p(-2,1) ) form a triangle. find the desired slopes and lengths, then fill in the words that characterize the triangle.
answer attempt 1 out of 3
slope of ( overline{no}= ) slope of ( overline{op}= ) slope of ( overline{np}=
length of ( overline{no}= ) length of ( overline{op}= ) length of ( overline{np}=
triangle nop is

Explanation:

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

Slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For \(N(0,-3)\) and \(O(3,4)\), \(m_{NO}=\frac{4-(-3)}{3 - 0}=\frac{7}{3}\)

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

For \(O(3,4)\) and \(P(-2,1)\), \(m_{OP}=\frac{1 - 4}{-2-3}=\frac{-3}{-5}=\frac{3}{5}\)

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

For \(N(0,-3)\) and \(P(-2,1)\), \(m_{NP}=\frac{1-(-3)}{-2 - 0}=\frac{4}{-2}=-2\)

Step4: Calculate length of \(\overline{NO}\)

Distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For \(N(0,-3)\) and \(O(3,4)\), \(d_{NO}=\sqrt{(3 - 0)^2+(4 + 3)^2}=\sqrt{9 + 49}=\sqrt{58}\)

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

For \(O(3,4)\) and \(P(-2,1)\), \(d_{OP}=\sqrt{(-2 - 3)^2+(1 - 4)^2}=\sqrt{25+9}=\sqrt{34}\)

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

For \(N(0,-3)\) and \(P(-2,1)\), \(d_{NP}=\sqrt{(-2 - 0)^2+(1 + 3)^2}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5}\)

Answer:

slope of \(\overline{NO}=\frac{7}{3}\), slope of \(\overline{OP}=\frac{3}{5}\), slope of \(\overline{NP}=-2\), length of \(\overline{NO}=\sqrt{58}\), length of \(\overline{OP}=\sqrt{34}\), length of \(\overline{NP}=2\sqrt{5}\)