Question

the coordinates of the vertices of △rst are r(-3,1), s(-1,4, and t(3,1)
which statement correctly describes whether △rst is a right triangle?
△rst is not a right triangle because no two sides are perpendicular
△rst is a right triangle because \overline{rs} is perpendicular to \overline{rt}
△rst is a right triangle because \overline{rs} is perpendicular to \overline{st}
△rst is a right triangle because \overline{st} is perpendicular to \overline{rt}

Explanation:

Step1: Calculate the slopes of the sides

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
For \(RS\): \(R(-3,1)\), \(S(-1,4)\)
\(m_{RS}=\frac{4 - 1}{-1-(-3)}=\frac{3}{2}\)
For \(RT\): \(R(-3,1)\), \(T(3,1)\)
\(m_{RT}=\frac{1 - 1}{3-(-3)} = 0\)
For \(ST\): \(S(-1,4)\), \(T(3,1)\)
\(m_{ST}=\frac{1 - 4}{3-(-1)}=-\frac{3}{4}\)

Step2: Check perpendicularity

Two lines with slopes \(m_1\) and \(m_2\) are perpendicular if \(m_1\times m_2=- 1\).
Since \(m_{RT} = 0\) (horizontal line) and \(m_{RS}=\frac{3}{2}
eq\infty\) (not vertical), \(RS\) and \(RT\) are not perpendicular.
\(m_{RS}\times m_{ST}=\frac{3}{2}\times(-\frac{3}{4})
eq - 1\), so \(RS\) and \(ST\) are not perpendicular.
Since \(m_{RT} = 0\) (horizontal line) and \(ST\) is not vertical (\(m_{ST}=-\frac{3}{4}\)), \(ST\) and \(RT\) are not perpendicular.

Answer:

\(\triangle RST\) is not a right triangle because no two sides are perpendicular.