Find the equation of the straight line passing through the point \$(3,2)\$ and perpendicular to the straight line joining the points \$(4,5)\$ and \$(1,2)\$.+

Line joining the points (4,5) and (1,2) and passing through the point \$(3,2)\$.

Slope of the straight line joining the points \$(4,5)\$ and \$(1,2)\$ is

\begin{align} \require{cancel} m_1 &= \frac{y_2 - y_1}{x_2 - x_1} \\ m_1 &= \frac{2 - 5}{1 - 4} \\ &= \frac{-3}{-3} \\ &= \frac{1}{1} \\ &= 1 \end{align}

If two straight lines are perpendicular then their slopes \$m_1 \times m_2 = -1\$.

\$∴\$ Slope of the required line is

\$m_2 = -1\$

The equation of a straight line perpendicular to the line joining the points (4,5) and (1,2) and passing through the point (3,2) is

\begin{align} \require{cancel} && y - y_1 &= m_2(x – x_1) \\ \Rightarrow && y - 2 &= - 1(x - 3) \\ \Rightarrow && y - 2 &= -x - 3 \\ \Rightarrow && x + y - 5 &= 0 \\ \end{align}

Answer

The equation of a straight line perpendicular to the line joining the points (4,5) and (1,2) and passing through the point (3,2) is \$ x + y - 5 = 0\$