Question

determine whether any of the lines in each figure are parallel or perpendicular.
directions: use slope to determine if lines pq and rs are parallel, perpendicular, or neither.

1. p(-9, -4), q(-7, -1), r(-2, 5), s(-6, -1)
2. p(-4, 17), q(1, -3), r(-9, 3), s(-5, 4)
3. p(-3, 14), q(2, -1), r(4, 8), s(-2, -10)
4. p(2, -1), q(-3, -1), r(-11, 9), s(-7, 9)

Explanation:

4.

Step1: Calculate the slope of \(\overrightarrow{PQ}\)

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For points \(P(-9,-4)\) and \(Q(-7,-1)\), \(m(\overrightarrow{PQ})=\frac{-1-(-4)}{-7 - (-9)}=\frac{-1 + 4}{-7+9}=\frac{3}{2}\)

Step2: Calculate the slope of \(\overrightarrow{RS}\)

For points \(R(-2,5)\) and \(S(-6,-1)\), \(m(\overrightarrow{RS})=\frac{-1 - 5}{-6-(-2)}=\frac{-6}{-4}=\frac{3}{2}\)

Step3: Determine the relationship

Since \(m(\overrightarrow{PQ})=m(\overrightarrow{RS})=\frac{3}{2}\), the lines are parallel.

Step1: Calculate the slope of \(\overrightarrow{PQ}\)

Using the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For points \(P(-4,17)\) and \(Q(1,-3)\), \(m(\overrightarrow{PQ})=\frac{-3 - 17}{1-(-4)}=\frac{-20}{5}=- 4\)

Step2: Calculate the slope of \(\overrightarrow{RS}\)

For points \(R(-9,3)\) and \(S(-5,4)\), \(m(\overrightarrow{RS})=\frac{4 - 3}{-5-(-9)}=\frac{1}{4}\)

Step3: Determine the relationship

Since \(m(\overrightarrow{PQ})\times m(\overrightarrow{RS})=-4\times\frac{1}{4}=-1\), the lines are perpendicular.

Step1: Calculate the slope of \(\overrightarrow{PQ}\)

Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For points \(P(-3,14)\) and \(Q(2,-1)\), \(m(\overrightarrow{PQ})=\frac{-1 - 14}{2-(-3)}=\frac{-15}{5}=-3\)

Step2: Calculate the slope of \(\overrightarrow{RS}\)

For points \(R(4,8)\) and \(S(-2,-10)\), \(m(\overrightarrow{RS})=\frac{-10 - 8}{-2 - 4}=\frac{-18}{-6}=3\)

Step3: Determine the relationship

Since \(m(\overrightarrow{PQ})
eq m(\overrightarrow{RS})\) and \(m(\overrightarrow{PQ})\times m(\overrightarrow{RS})
eq - 1\), the lines are neither parallel nor perpendicular.

Answer:

\(m(\overrightarrow{PQ})=\frac{3}{2}\), \(m(\overrightarrow{RS})=\frac{3}{2}\), Types of Lines: Parallel

5.