Which lines are perpendicular to the line [tex]y-1=\frac{1}{3}(x+2)[/tex]? Check all that apply.
[tex]y+2=-3(x-4)[/tex]
[tex]y-5=3(x+11)[/tex]
[tex]y=-3 x-\frac{5}{3}[/tex]
[tex]y=\frac{1}{3} x-2[/tex]
[tex]3 x+y=7[/tex]
Answer (2)
Find the slope of the given line y − 1 = 3 1 ( x + 2 ) , which is 3 1 .
Identify lines with a slope of -3, since the product of the slopes of perpendicular lines is -1.
Check each line to see if its slope is -3.
The lines y + 2 = − 3 ( x − 4 ) , y = − 3 x − 3 5 , and 3 x + y = 7 are perpendicular to the given line. Therefore, the answer is y + 2 = − 3 ( x − 4 ) , y = − 3 x − 3 5 , 3 x + y = 7 .
Explanation
Problem Analysis We are given the line y − 1 = 3 1 ( x + 2 ) and asked to find which of the following lines are perpendicular to it:
y + 2 = − 3 ( x − 4 )
y − 5 = 3 ( x + 11 )
y = − 3 x − 3 5
y = 3 1 x − 2
3 x + y = 7
Two lines are perpendicular if the product of their slopes is -1.
Finding the Slope of the Given Line First, let's find the slope of the given line. We can rewrite the equation in slope-intercept form, y = m x + b , where m is the slope. The given line is y − 1 = 3 1 ( x + 2 ) . Adding 1 to both sides gives y = 3 1 ( x + 2 ) + 1 = 3 1 x + 3 2 + 1 = 3 1 x + 3 5 . So the slope of the given line is 3 1 .
Checking Each Line for Perpendicularity Now, we need to find lines with a slope of -3, since 3 1 × − 3 = − 1 . Let's examine each of the given lines.
Line 1: y + 2 = − 3 ( x − 4 ) . This can be rewritten as y = − 3 ( x − 4 ) − 2 = − 3 x + 12 − 2 = − 3 x + 10 . The slope is -3, so this line is perpendicular to the given line.
Line 2: y − 5 = 3 ( x + 11 ) . This can be rewritten as y = 3 ( x + 11 ) + 5 = 3 x + 33 + 5 = 3 x + 38 . The slope is 3, so this line is not perpendicular to the given line.
Line 3: y = − 3 x − 3 5 . The slope is -3, so this line is perpendicular to the given line.
Line 4: y = 3 1 x − 2 . The slope is 3 1 , so this line is not perpendicular to the given line.
Line 5: 3 x + y = 7 . This can be rewritten as y = − 3 x + 7 . The slope is -3, so this line is perpendicular to the given line.
Final Answer Therefore, the lines that are perpendicular to the given line are:
y + 2 = − 3 ( x − 4 )
y = − 3 x − 3 5
3 x + y = 7
Examples
Understanding perpendicular lines is crucial in many real-world applications, such as architecture and construction. For example, when building a house, the walls need to be perpendicular to the floor to ensure stability. Similarly, in navigation, understanding perpendicular paths is essential for determining the shortest distance between two points or for avoiding obstacles. This concept also applies to computer graphics, where perpendicular vectors are used to create realistic lighting and shading effects.
The lines that are perpendicular to the given line are: y + 2 = − 3 ( x − 4 ) , y = − 3 x − 3 5 , and 3 x + y = 7 .
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