Question

What is the slope of a line that is perpendicular to the line that passes through (-6, 6) and (-2, -13)?

Answer 1

Hint: The slope of a line, also known as the gradient of a line, is a numerical value that indicates the line's direction and steepness. If two coordinates ( \[{x_1}\] , \[{y_1}\] ) and ( \[{x_2}\] , \[{y_2}\] ) are given, the slope of a line passing through these locations is: \[slope = \left( {\dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}} \right)\] . The negative inverse of the slope of a line perpendicular to a given line is the slope of the provided line.

Complete answer:
We have given the points (3, -2) and (3, 4) through which the line passes. The slope of a line going between the coordinates ( \[{x_1}\] , \[{y_1}\] ) and ( \[{x_2}\] , \[{y_2}\] ) may now be calculated as follows:
\[slope = \left( {\dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}} \right)\]
We will substitute \[{x_1} = - 6\] , \[{y_1} = 6\] , \[{x_2} = - 2\] , \[{y_2} = - 13\] in the equation
\[ \Rightarrow slope = \left( {\dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}} \right)\]
\[ \Rightarrow slope = \left( {\dfrac{{ - 13 - 6}}{{ - 2 - ( - 6)}}} \right)\]
\[ \Rightarrow slope = \left( {\dfrac{{ - 19}}{4}} \right)\]
We know that the negative inverse of the slope of a line perpendicular to a given line is the slope of the provided line.
So, if the slope of a line is m,
The slope of perpendicular line is \[ - \dfrac{1}{m}\]
We have given the slope of line \[\dfrac{{ - 19}}{4}\]
The slope of perpendicular line is \[\dfrac{4}{{19}}\]
Hence, the slope of a line that is perpendicular to the line that passes through (-6, 6) and (-2, -13) is \[\dfrac{4}{{19}}\]

Note: Remember that a line is parallel to the x-axis if its slope is zero, and it is perpendicular to the x-axis if its slope goes to infinity. You should also keep in mind that if the x-coordinates of the two points through which the line travels are the same, the line must be perpendicular to the x-axis, and if the y-coordinates of the two locations through which the line passes are the same, the line must be parallel to the y-axis.