Question

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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 line that passes through 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 the same slope.
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 the point (-6,5)? show all of your work.
2. what is the equation of the line:

• parallel to the line y = -1/4x + 5 and
• passing through the point (2, -1)

Explanation:

Step1: Recall slope - perpendicular/parallel rules

Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals (opposite reciprocals) of each other. The equation of a line is $y = mx + b$, where $m$ is the slope.

Step2: Rewrite given line for question 1 in slope - intercept form

Rewrite $2x - y=4$ as $y = 2x - 4$, so its slope $m_1 = 2$. A perpendicular line has slope $m_2=-\frac{1}{2}$. The answer is c) $y =-\frac{1}{2}x - 12$.

Step3: Calculate slopes for question 2

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For line A with points $(-8,5)$ and $(-5,4)$, $m_A=\frac{4 - 5}{-5+8}=-\frac{1}{3}$. For line B with points $(0,1)$ and $(4,-1)$, $m_B=\frac{-1 - 1}{4 - 0}=-\frac{1}{2}$. Since the slopes are not equal and not negative reciprocals, lines A and B intersect, so the answer is d).

Step4: Find parallel line for question 3

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

Step5: Find perpendicular line for question 4

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

Step6: Find parallel line for question 5

The line $y=-\frac{1}{4}x + 5$ has slope $m=-\frac{1}{4}$. Using the point - slope form $y - y_1=m(x - x_1)$ with point $(2,-1)$ and $m =-\frac{1}{4}$, we have $y+1=-\frac{1}{4}(x - 2)$. Expand: $y + 1=-\frac{1}{4}x+\frac{1}{2}$, so $y=-\frac{1}{4}x-\frac{1}{2}$.

Answer:

1. c) $y =-\frac{1}{2}x - 12$
2. d) Lines A and B intersect, because their slopes have no relationship.
3. b) $y = 2x-15$
4. $y=-\frac{1}{2}x + 2$
5. $y=-\frac{1}{4}x-\frac{1}{2}$