What is the equation, in slope-intercept form, of the line that is perpendicular to the line [tex]y-4=-\frac{2}{3}(x-6)[/tex] and passes through the point $(-2,-2)$?
A. [tex]y=-\frac{2}{3} x-\frac{10}{3}[/tex]
B. [tex]y=-\frac{2}{3} x+\frac{10}{3}[/tex]
C. [tex]y=\frac{3}{2} x-1[/tex]
D. [tex]y=\frac{3}{2} x+1[/tex]
Answer (2)
Find the slope of the given line: m 1 = − 3 2 .
Find the slope of the perpendicular line: m 2 = 2 3 .
Use the point-slope form with the point ( − 2 , − 2 ) and slope 2 3 : y + 2 = 2 3 ( x + 2 ) .
Convert to slope-intercept form: y = 2 3 x + 1 .
y = 2 3 x + 1
Explanation
Find the slope of the given line The given equation is y − 4 = − 3 2 ( x − 6 ) . First, we need to find the slope of this line. The equation is in point-slope form, y − y 1 = m ( x − x 1 ) , where m is the slope. In this case, the slope of the given line is − 3 2 .
Find the slope of the perpendicular line The line we want to find is perpendicular to the given line. The slope of a perpendicular line is the negative reciprocal of the original line's slope. So, the slope of the perpendicular line is the negative reciprocal of − 3 2 , which is 2 3 .
Use the point-slope form of a line Now we know the slope of the perpendicular line is 2 3 , and it passes through the point ( − 2 , − 2 ) . We can use the point-slope form of a line, y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is the given point and m is the slope of the perpendicular line. Substituting the point ( − 2 , − 2 ) and the slope 2 3 into the point-slope form, we get:
y − ( − 2 ) = 2 3 ( x − ( − 2 ))
y + 2 = 2 3 ( x + 2 )
Convert to slope-intercept form Now, we need to convert the equation to slope-intercept form, y = m x + b .
y + 2 = 2 3 ( x + 2 )
y + 2 = 2 3 x + 2 3 ( 2 )
y + 2 = 2 3 x + 3
y = 2 3 x + 3 − 2
y = 2 3 x + 1
Final Answer The equation of the line in slope-intercept form that is perpendicular to the line y − 4 = − 3 2 ( x − 6 ) and passes through the point ( − 2 , − 2 ) is y = 2 3 x + 1 .
Examples
Understanding perpendicular lines is crucial in architecture and construction. For instance, when designing a building, ensuring walls are perpendicular to the ground is essential for stability. If a surveyor determines a wall's slope needs adjustment, they use the concept of perpendicular slopes to calculate the precise angle for correction, ensuring the structure's integrity and safety.
The equation of the line perpendicular to the given line and passing through the point (-2, -2) is y = 2 3 x + 1 . Thus, the answer is option D. This involves finding the slope of the original line, determining the negative reciprocal for the perpendicular slope, and using point-slope form to derive the equation.
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