Question

Line A passes through the points (0,2) and (2,−2). Line B passes through the points (0,4) and (1,2). Line C passes through the points (0,−1) and (4,1). Identify which lines are parallel and which lines are perpendicular.

Answer 1

A || B, A ⊥ C, B ⊥ C

Explanation:

`\text{Let}\ k:y=m_1x+b_1\ \text{and}\ l:y=m_2x+b_2\\\\l\ \perp\ k\iff m_1m_2=-1\to m_2=-(1)/(m_1)\\\\l\ \parallel\ k\iff m_1=m_2\\\\m_1,\ m_2-\ slope\\b_1,\ b_2-\ y-intercept\\\\\text{The formula of a slope:}\\\\m=(y_2-y_1)/(x_2-x_1)\\-----------------------`

`\text{We have points through which the line passes}\\\\Line\ A:\ (0,\ 2),\ (2,\ -2).\\Line\ B:\ (0,\ 4),\ (1,\ 2).\\Line\ C:\ (0,\ -1),\ (4,\ 1).\\\\\text{Calculte the slopes:}`

`m_A=(-2-2)/(2-0)=(-4)/(2)=-2\\\\m_B=(2-4)/(1-0)=(-2)/(1)=-2\\\\m_C=(1-(-1))/(4-0)=(2)/(4)=(1)/(2)\\\\m_A=m_B\to\text{therefore the lines A and B are parallel}\\\\\begin{array}{ccc}m_Am_C=-2\cdot(1)/(2)=-1\\\\m_Bm_C=-2\cdot(1)/(2)=-1\end{array}\to\begin{array}{ccc}\text{therefore the lines A and C}\\\text{and B and C are perpendicular}\end{array}`