Question

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the coordinates of the vertices of △jkl are j(-5, -1), k(0, 1), and l(2, -5).
which statement correctly describes whether △jkl is a right triangle?
○ △jkl is a right triangle because (overline{jk}) is perpendicular to (overline{kl}).
○ △jkl is a right triangle because (overline{jl}) is perpendicular to (overline{kl}).
○ △jkl is not a right triangle because no two of its sides are perpendicular.
○ △jkl is a right triangle because (overline{jk}) is perpendicular to (overline{jl}).

Explanation:

Step1: Calculate the slope of \(JK\)

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For points \(J(-5,-1)\) and \(K(0,1)\), \(m_{JK}=\frac{1 - (-1)}{0 - (-5)}=\frac{2}{5}\)

Step2: Calculate the slope of \(KL\)

For points \(K(0,1)\) and \(L(2,-5)\), \(m_{KL}=\frac{-5 - 1}{2 - 0}=\frac{-6}{2}=-3\)

Step3: Calculate the slope of \(JL\)

For points \(J(-5,-1)\) and \(L(2,-5)\), \(m_{JL}=\frac{-5 - (-1)}{2 - (-5)}=\frac{-4}{7}\)

Step4: Check perpendicularity

Two lines are perpendicular if \(m_1\times m_2=-1\).
\(m_{JK}\times m_{KL}=\frac{2}{5}\times(-3)=-\frac{6}{5}
eq - 1\)
\(m_{JL}\times m_{KL}=\frac{-4}{7}\times(-3)=\frac{12}{7}
eq - 1\)
\(m_{JK}\times m_{JL}=\frac{2}{5}\times\frac{-4}{7}=-\frac{8}{35}
eq - 1\)

Answer:

\(\triangle JKL\) is not a right triangle because no two of its sides are perpendicular.