Question

find the desired slopes and lengths, then fill in the words that characterize the triangle.
answer attempt 1 out of 2
s l o p e o f p overline { q } =
length of overline { p q } =
s l o p e o f q overline { r } =
length of overline { q r } =
s l o p e o f r overline { p } =
length of overline { r p } =
triangle pqr is

Explanation:

Step1: Find coordinates of points

Assume \(P=(0, - 3)\), \(Q=(6, - 2)\), \(R=(3,1)\)

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

Use slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
For \(P(0,-3)\) and \(Q(6,-2)\), \(m_{PQ}=\frac{-2-(-3)}{6 - 0}=\frac{-2 + 3}{6}=\frac{1}{6}\)

Step3: Calculate length of \(\overline{PQ}\)

Use distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)
\(d_{PQ}=\sqrt{(6 - 0)^2+(-2+3)^2}=\sqrt{36 + 1}=\sqrt{37}\)

Step4: Calculate slope of \(\overline{QR}\)

For \(Q(6,-2)\) and \(R(3,1)\), \(m_{QR}=\frac{1-(-2)}{3 - 6}=\frac{1 + 2}{-3}=-1\)

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

\(d_{QR}=\sqrt{(3 - 6)^2+(1 + 2)^2}=\sqrt{9+9}=\sqrt{18}=3\sqrt{2}\)

Step6: Calculate slope of \(\overline{RP}\)

For \(R(3,1)\) and \(P(0,-3)\), \(m_{RP}=\frac{-3-1}{0 - 3}=\frac{-4}{-3}=\frac{4}{3}\)

Step7: Calculate length of \(\overline{RP}\)

\(d_{RP}=\sqrt{(0 - 3)^2+(-3 - 1)^2}=\sqrt{9 + 16}=\sqrt{25}=5\)

Since no two sides are equal and no slope of two sides is \(m_1\times m_2=-1\) (no right - angle), triangle \(PQR\) is scalene.

Answer:

slope of \(\overline{PQ}=\frac{1}{6}\), length of \(\overline{PQ}=\sqrt{37}\), slope of \(\overline{QR}=-1\), length of \(\overline{QR}=3\sqrt{2}\), slope of \(\overline{RP}=\frac{4}{3}\), length of \(\overline{RP}=5\), Triangle \(PQR\) is scalene.