Question

which of the lines are perpendicular?
□ a and d
□ b and d
□ c and d
which of the lines are parallel?
□ a and b
□ a and c
□ b and c

Explanation:

Step1: Calculate slope of line a

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For line a with points $(-1,4)$ and $(5,6)$, $m_a=\frac{6 - 4}{5-(-1)}=\frac{2}{6}=\frac{1}{3}$.

Step2: Calculate slope of line b

For line b with points $(-1,1)$ and $(3,2)$, $m_b=\frac{2 - 1}{3-(-1)}=\frac{1}{4}$.

Step3: Calculate slope of line c

For line c with points $(-3,-2)$ and $(3,0)$, $m_c=\frac{0 - (-2)}{3-(-3)}=\frac{2}{6}=\frac{1}{3}$.

Step4: Calculate slope of line d

For line d with points $(1,6)$ and $(3,-2)$, $m_d=\frac{-2 - 6}{3 - 1}=\frac{-8}{2}=-4$.

Step5: Check perpendicularity

Two lines are perpendicular if the product of their slopes is - 1. $m_b\times m_d=\frac{1}{4}\times(-4)=-1$, so b and d are perpendicular.

Step6: Check parallelism

Two lines are parallel if they have the same slope. $m_a = m_c=\frac{1}{3}$, so a and c are parallel.

Answer:

b and d are perpendicular; a and c are parallel. So the answers are:

• For perpendicular lines: b and d
• For parallel lines: a and c