Question

lines mn and pq are parallel. lines rs and tv intersect them. which statements are true about these lines? choose three correct answers the slope of line rs is $-\frac{3}{2}$. line rs is perpendicular to both line mn and line pq. the slope of line pq is undefined. the slope of line mn is $\frac{2}{3}$. lines rs and tv are parallel.

Explanation:

Step1: Calculate slope of line RS

Let \(R(-1,4)\) and \(S(2,-2)\).
Slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
\(m_{RS}=\frac{-2 - 4}{2-(-1)}=\frac{-6}{3}=-\frac{3}{2}\).

Step2: Calculate slope of line MN

Let \(M(-3,-1)\) and \(N(3,3)\).
\(m_{MN}=\frac{3-(-1)}{3-(-3)}=\frac{4}{6}=\frac{2}{3}\).

Step3: Check parallel and perpendicular

• For perpendicular lines \(m_1\times m_2=-1\). \(m_{RS}\times m_{MN}=-\frac{3}{2}\times\frac{2}{3}=-1\). But line \(RS\) is not perpendicular to \(PQ\) (since \(PQ\) is not vertical or horizontal in a way to satisfy perpendicularity with \(RS\) as calculated).
• Line \(PQ\) is not vertical (so slope is not undefined).
• For parallel lines \(m_1 = m_2\). Slope of \(TV\) (let \(T(-4,1)\) and \(V(0,-4)\), \(m_{TV}=\frac{-4 - 1}{0-(-4)}=-\frac{5}{4}

eq m_{RS}\)).

Answer:

The slope of line \(RS\) is \(-\frac{3}{2}\), The slope of line \(MN\) is \(\frac{2}{3}\)