1.2: Perpendicular lines and an isosceles triangle

The straight line $\displaystyle L_1$ passes through the point $\displaystyle A$ with coordinates $\displaystyle (4,7)$ and has gradient $\displaystyle m$, where $\displaystyle m<0$.

Another straight line $\displaystyle L_2$ is perpendicular to $\displaystyle L_1$ and passes through the point $\displaystyle B$ with coordinates $\displaystyle (4,k)$, where $\displaystyle k\neq7$.

The lines $\displaystyle L_1$ and $\displaystyle L_2$ intersect at the point $\displaystyle C$. Given that the $\displaystyle y$ coordinate of $\displaystyle C$ is $\displaystyle Y$,

(a) show that

Given that the triangle $\displaystyle ABC$ is isosceles,

(b) find the value of $\displaystyle m$.