Question

graph 1: points c(-3,2), d(3,-1), a(8,-2), b(5,-8); options parallel, perpendicular, neither. graph 2: points a(-1,5), b(2,2), c(1,8), d(6,3); options parallel, perpendicular, neither. graph 3: points a(2,-4), b(1,-6), c(4,-8), d(3,-3); options parallel, perpendicular, neither.

Explanation:

Step1: Analyze Graph 2

First, we find the slopes of the two lines in Graph 2. For the line passing through points \( A(-1, 5) \) and \( B(2, 2) \), the slope \( m_1 \) is calculated using the slope formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \). So, \( m_1=\frac{2 - 5}{2 - (-1)}=\frac{-3}{3}=- 1 \)? Wait, no, wait. Wait, points \( A(-1,5) \), \( B(2,2) \): \( y_2 - y_1=2 - 5=-3 \), \( x_2 - x_1=2 - (-1)=3 \), so \( m_1=\frac{-3}{3}=-1 \)? Wait, no, wait the other line: points \( C(1,8) \) and \( D(6,3) \). \( m_2=\frac{3 - 8}{6 - 1}=\frac{-5}{5}=-1 \). Wait, no, that can't be. Wait, maybe I misread the points. Wait, Graph 2: points \( A(-1,5) \), \( B(2,2) \); \( C(1,8) \), \( D(6,3) \). Wait, let's recalculate \( m_1 \): \( (2 - 5)/(2 - (-1))=-3/3=-1 \). \( m_2=(3 - 8)/(6 - 1)=-5/5=-1 \). So both slopes are equal, so the lines are parallel. Wait, but let's check another graph. Wait, maybe the question is about which graph has parallel lines. Let's check Graph 2 again. The two lines: one through \( A(-1,5) \) and \( B(2,2) \), another through \( C(1,8) \) and \( D(6,3) \). Slope of first line: \( (2 - 5)/(2 - (-1))=-3/3=-1 \). Slope of second line: \( (3 - 8)/(6 - 1)=-5/5=-1 \). So slopes are equal, so lines are parallel. So Graph 2 has parallel lines.

Step2: Confirm

In Graph 1, let's check slopes. Points \( C(-3,2) \) and \( D(3,-1) \): slope \( m_1=\frac{-1 - 2}{3 - (-3)}=\frac{-3}{6}=-\frac{1}{2} \). Points \( A(8,-2) \) and \( B(5,-8) \): slope \( m_2=\frac{-8 - (-2)}{5 - 8}=\frac{-6}{-3}=2 \). \( m_1\times m_2=-\frac{1}{2}\times2=-1 \), so they are perpendicular. In Graph 3, let's check slopes. Points \( A(2,-4) \) and \( C(7,-8) \): slope \( m_1=\frac{-8 - (-4)}{7 - 2}=\frac{-4}{5} \). Points \( B(1,-6) \) and \( D(3,-3) \): slope \( m_2=\frac{-3 - (-6)}{3 - 1}=\frac{3}{2} \). \( m_1\times m_2=\frac{-4}{5}\times\frac{3}{2}=-\frac{6}{5}
eq - 1 \), and \( m_1
eq m_2 \), so neither. So Graph 2 has parallel lines.

Answer:

Graph 2 (the middle graph) has parallel lines, so the correct option is the one corresponding to Graph 2 (the middle graph) with the "parallel" radio button selected. But since the question is about which graph shows parallel lines, the answer is Graph 2 (the middle graph).