Given:
The line through (6, −3) and (7, 6) is perpendicular to the line through (−2, 5) and (x, −1).
To find:
The value of x.
Solution:
Slope formula: If a line passes through two points, then the slope of the line is:
[tex]m=\dfrac{y_2-y_1}{x_2-x_1}[/tex]
The slope of the line through (6, −3) and (7, 6) is:
[tex]m_1=\dfrac{6-(-3)}{7-6}[/tex]
[tex]m_1=\dfrac{6+3}{1}[/tex]
[tex]m_1=9[/tex]
The slope of the line through (−2, 5) and (x, −1) is:
[tex]m_2=\dfrac{-1-5}{x-(-2)}[/tex]
[tex]m_2=\dfrac{-6}{x+2}[/tex]
We know that the product of slopes of two perpendicular lines is -1.
[tex]m_1\cdot m_2=-1[/tex]
[tex]9\cdot \dfrac{-6}{x+2}=-1[/tex]
[tex]\dfrac{-54}{x+2}=-1[/tex]
Multiplying both sides by (x+2), we get
[tex]-54=-1(x+2)[/tex]
[tex]-54=-x-2[/tex]
[tex]-54+2=-x[/tex]
[tex]-52=-x[/tex]
Divide both sides by -1.
[tex]52=x[/tex]
Therefore, the value of x is 52.
