Question

name:
er.dfa1.geo.m.g.gpe.b.05
prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems including finding the equation of a line parallel or perpendicular to a given point.

1. which of the following lines is perpendicular to the line 2x - y = 4?

a) y = -2x - 12
b) y = 1/2x + 12
c) y = -1/2x - 12
d) y = 2x + 12

1. line a passes through the points (-8, 5) and (-5, 4). line b passes through the points (0, 1) and (4, -1). which of the following describes the relationship between line a and line b?

a) lines a and b are parallel, because they have the same slope.
b) lines a and b are parallel, because they have opposite reciprocal slopes.
c) lines a and b are perpendicular, because they have opposite reciprocal slopes.
d) lines a and b intersect, because their slopes have no relationship.

1. which equation is parallel to the line y = 2x+6 and passes through the point (8,1)?

a) y = 2x+6
b) y = 2x - 15
c) y = 1/2x + 10
d) y = -2x - 7

1. what is the equation that is perpendicular to the line y = 2x - 3 and passes through point (-6,5)? show all of your work.

Explanation:

Step1: Recall slope - related rules

For two lines $y = m_1x + b_1$ and $y=m_2x + b_2$, if they are parallel, $m_1=m_2$; if they are perpendicular, $m_1\times m_2=- 1$.

Step2: Solve problem 1

Rewrite the line $2x - y=4$ as $y = 2x-4$, its slope $m = 2$. The slope of a perpendicular line is $-\frac{1}{2}$. Among the options, the line $y=-\frac{1}{2}x - 12$ has a slope of $-\frac{1}{2}$, so the answer is c.

Step3: Solve problem 2

The slope of line A passing through $(-8,5)$ and $(-5,4)$ is $m_A=\frac{4 - 5}{-5+8}=-\frac{1}{3}$. The slope of line B passing through $(0,1)$ and $(4,-1)$ is $m_B=\frac{-1 - 1}{4-0}=-\frac{1}{2}$. Since $m_A
eq m_B$ and $m_A\times m_B
eq - 1$, lines A and B intersect and have no special relationship, so the answer is d.

Step4: Solve problem 3

The line parallel to $y = 2x+6$ has a slope $m = 2$. Using the point - slope form $y - y_1=m(x - x_1)$ with $(x_1,y_1)=(8,1)$ and $m = 2$, we have $y - 1=2(x - 8)$. Expand it: $y-1=2x-16$, so $y = 2x-15$. The answer is b.

Step5: Solve problem 4

The slope of the line $y = 2x-3$ is $m_1 = 2$. The slope of a perpendicular line $m_2=-\frac{1}{2}$. Using the point - slope form $y - y_1=m_2(x - x_1)$ with $(x_1,y_1)=(-6,5)$ and $m_2=-\frac{1}{2}$, we get $y - 5=-\frac{1}{2}(x + 6)$. Expand it: $y-5=-\frac{1}{2}x-3$, then $y=-\frac{1}{2}x + 2$.

Answer:

1. c
2. d
3. b
4. $y=-\frac{1}{2}x + 2$