Write an equation of the line that is perpendicular to the line through 9,10 and 3,-2 and passes through the x-intercept of that line.

Write an equation of the line that is perpendicular to the line through 9,10 and 3,-2 and passes through the x-intercept of that line.

\(\small{ \begin{array}{rcll} \hline \text{line } 1\quad m(\text{slope})=\ ?\\ m &=& \dfrac{y_2-y_1}{x_2-x_1} \qquad ( x_1 = 3,\ y_1=-2 ) \quad ( x_2=9,\ y_2 = 10 )\\ m &=& \dfrac{10-(-2)}{9-3} \\ m &=& \dfrac{12}{6} \\ \mathbf{m} &\mathbf{=}& \mathbf{2} \\\\ m_{\text{perpendicular}} &=& -\dfrac{1}{m} \qquad m=2\\ \mathbf{m_{\text{perpendicular}}} &\mathbf{=}& \mathbf{-\dfrac{1}{2}} \\ \hline \text{line } 1\quad x_{(intercept)}=x_i=\ ?\\ m &=& \dfrac{y-y_1}{x-x_1} \\ m &=& \dfrac{y_i-y_1}{x_i-x_1} \qquad ( x_1 = 3,\ y_1=-2 ) \quad m=2 \quad y_i = 0\\ 2 &=& \dfrac{0-(-2)}{x_i-3} \\ 2 &=& \dfrac{0-(-2)}{x_i-3} \\ 2 &=& \dfrac{2}{x_i-3} \qquad & | \qquad :2 \\ 1 &=& \dfrac{1}{x_i-3} \\ x_i-3 &=& 1 \\ x_i &=& 1+3 \\ x_i &=& 4 \\ \mathbf{\text{Point x-intercept}:\ ( x_i = 4,\ y_i=0 )}\\ \hline \text{line perpendicular:} \\ m_{\text{perpendicular}} &=& \dfrac{y-y_i}{x-x_i} \qquad ( x_i = 4,\ y_i=0 ) \quad m_{\text{perpendicular}} =-\dfrac12 \\ -\dfrac12 &=& \dfrac{y-0}{x-4} \\ -\dfrac12 &=& \dfrac{y}{x-4} \\ y &=& -\dfrac12 \cdot (x-4) \\ y &=& -\dfrac12 \cdot x - ( -\dfrac12 )(4) \\ \mathbf{y} &\mathbf{=}& \mathbf{-\dfrac12 \cdot x +2} \\ \end{array} }\)