The equation of line [tex]$CD$[/tex] is [tex]$y=3x-3$[/tex]. Write an equation of a line perpendicular to line [tex]$CD$[/tex] in slope-intercept form that contains point [tex]$(3,1)$[/tex].
Answer (2)
Determine the slope of the given line: The slope of y = 3 x − 3 is 3 .
Calculate the slope of the perpendicular line: The negative reciprocal of 3 is − 3 1 .
Use the point-slope form with the point ( 3 , 1 ) and slope − 3 1 : y − 1 = − 3 1 ( x − 3 ) .
Convert to slope-intercept form: y = − 3 1 x + 2 . The final answer is y = 3 − 1 x + 2 .
Explanation
Understanding the Problem The equation of line C D is given as y = 3 x − 3 . We need to find the equation of a line perpendicular to this line that passes through the point ( 3 , 1 ) . The final equation should be in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept.
Finding the Perpendicular Slope The slope of the given line C D is 3 . The slope of a line perpendicular to line C D is the negative reciprocal of the slope of C D . Therefore, the slope of the perpendicular line is − 3 1 .
Using Point-Slope Form Now we use the point-slope form of a line, which is y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is the given point and m is the slope. We have the point ( 3 , 1 ) and the slope − 3 1 . Plugging these values into the point-slope form, we get:
y − 1 = − 3 1 ( x − 3 )
Converting to Slope-Intercept Form Now, we convert the equation to slope-intercept form ( y = m x + b ).
y − 1 = − 3 1 x + 1
y = − 3 1 x + 1 + 1
y = − 3 1 x + 2
Final Answer The equation of the line perpendicular to y = 3 x − 3 and passing through the point ( 3 , 1 ) is y = − 3 1 x + 2 .
Examples
Imagine you're designing a rectangular garden and one side needs to be perpendicular to an existing fence. If the fence's line equation is y = 3 x − 3 , you can use the principles of perpendicular lines to determine the slope and equation for the garden's side. Knowing the coordinates of one corner of the garden, say ( 3 , 1 ) , allows you to define the exact equation of the garden's side, ensuring it's perfectly perpendicular to the fence. This ensures your garden design is geometrically sound and aesthetically pleasing.
To find the equation of a line perpendicular to y = 3 x − 3 and passing through ( 3 , 1 ) , we first determine the slope of the perpendicular line as − 3 1 . Using the point-slope form, we find that the equation is y = − 3 1 x + 2 .
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