1.5: Coordinates, perpendicular lines and area

The point $\displaystyle A$ has coordinates $\displaystyle (-4,-10)$ and the point $\displaystyle B$ has coordinates $\displaystyle (3,11)$. The line $\displaystyle l$ passes through $\displaystyle A$ and $\displaystyle B$.

(a) Find an equation of $\displaystyle l$.

The point $\displaystyle P$ lies on $\displaystyle l$ such that $\displaystyle AP:PB=3:4$.

(b) Find the coordinates of $\displaystyle P$.

The point $\displaystyle Q$ with coordinates $\displaystyle (m,n)$, where $\displaystyle m<0$, lies on the line through $\displaystyle P$ that is perpendicular to $\displaystyle l$. Given that the length of $\displaystyle PQ$ is $\displaystyle \sqrt{10}$,

(c) find the coordinates of $\displaystyle Q$.

The point $\displaystyle R$ has coordinates $\displaystyle (-11,-21)$.

(d) Show that

(i) $\displaystyle AB$ and $\displaystyle RQ$ are equal in length,

(ii) $\displaystyle AB$ and $\displaystyle RQ$ are parallel.

(e) Find the area of the quadrilateral $\displaystyle ABQR$.