1.27: Coordinate Geometry and a Cyclic Trapezium

The points $\displaystyle A$ and $\displaystyle B$ have coordinates $\displaystyle (-1,3)$ and $\displaystyle (5,6)$ respectively.

(a) Find an equation for the line $\displaystyle AB$.

The point $\displaystyle P$ divides $\displaystyle AB$ in the ratio $\displaystyle 2:1$.

(b) Show that the coordinates of $\displaystyle P$ are $\displaystyle (3,5)$.

The point $\displaystyle C$ with coordinates $\displaystyle (m,n)$, where $\displaystyle m>0$, is such that $\displaystyle CP$ is perpendicular to the line $\displaystyle AB$.

Given that the radius of the circle which passes through $\displaystyle A$, $\displaystyle P$ and $\displaystyle C$ is 5,

(c) find the value of $\displaystyle m$ and the value of $\displaystyle n$.

The point $\displaystyle D$ with coordinates $\displaystyle (p,q)$ is such that the line $\displaystyle AD$ is perpendicular to the line $\displaystyle AB$ and the line $\displaystyle DC$ is parallel to the line $\displaystyle AB$.

(d) Find the value of $\displaystyle p$ and the value of $\displaystyle q$.

(e) Find the area of trapezium $\displaystyle ABCD$.