Find the equation of the line perpendicular to [tex]y=-\frac{1}{3} x+1[/tex] that includes the point (1,2).
Give your answer in Point-Slope Form.
[tex]y-2=[?](x-\square)[/tex]
Point-Slope Form: [tex]y-y_1=m\left(x_1-x_1\right)[/tex]
Answer (2)
Find the slope of the perpendicular line by taking the negative reciprocal of the given line's slope: m = 3 .
Use the point-slope form of a line, y − y 1 = m ( x − x 1 ) , with the point ( 1 , 2 ) and the slope m = 3 .
Substitute the point ( 1 , 2 ) and the slope m = 3 into the point-slope form: y − 2 = 3 ( x − 1 ) .
The equation of the line is y − 2 = 3 ( x − 1 ) .
Explanation
Understanding the Problem We are given the equation of a line y = − 3 1 x + 1 and a point ( 1 , 2 ) . We need to find the equation of the line that is perpendicular to the given line and passes through the given point. The equation should be in point-slope form, which is y − y 1 = m ( x − x 1 ) , where m is the slope and ( x 1 , y 1 ) is a point on the line.
Finding the Slope of the Perpendicular Line The slope of the given line is − 3 1 . The slope of a line perpendicular to this line is the negative reciprocal of − 3 1 , which is 3 .
Using Point-Slope Form Now we use the point-slope form y − y 1 = m ( x − x 1 ) with the point ( 1 , 2 ) and the slope m = 3 . Substituting these values, we get y − 2 = 3 ( x − 1 ) .
Final Answer The equation of the line perpendicular to y = − 3 1 x + 1 that passes through the point ( 1 , 2 ) in point-slope form is y − 2 = 3 ( x − 1 ) . Therefore, the missing value in y − 2 = [ ?] ( x − □ ) is 3 and 1 respectively.
Examples
Imagine you're designing a rectangular garden and one side must align perpendicular to an existing fence represented by the line y = − 3 1 x + 1 . If you want a corner of your garden to be at the point (1, 2), finding the equation of the line that forms the garden's side helps you plan the garden's layout. This ensures the garden side is perfectly perpendicular to the fence, optimizing space and aesthetics. Understanding perpendicular lines is crucial in design, construction, and even navigation, ensuring accuracy and stability in various real-world applications.
The equation of the line perpendicular to y = − 3 1 x + 1 that passes through the point ( 1 , 2 ) is y − 2 = 3 ( x − 1 ) . This was found by identifying the slope of the given line, calculating its negative reciprocal, and using the point-slope form of a line.
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