The figure shows a right-angled triangle $ABC$ in which $A$, $B$ and $C$ are $(-3,\ 18)$,

Question

The figure shows a right-angled triangle $ABC$ in which $A$, $B$ and $C$ are $(-3,\ 18)$, $(k,\ 0)$ and $(15,\ 4)$ respectively. $AB$ cuts the $y$-axis at $P$. The point $Q$ lies on $AC$ and $PQ$ is parallel to $BC$. Given that $k<5$, find the value of $k$.[3] Find the coordinates of $P$ and show that $P$ is the midpoint of $AB$.[3] Hence find the coordinates of $Q$.[2] The point $D$ is such that $ABCD$ is a rectangle. Find the coordinates of $D$.[2] Find the area of the rectangle $ABCD$.[3] The point $F$ lies on $BC$ such that $BF : FC = 1 : 3$. Find the coordinates of $F$.[2]

Accepted Answer

(a) $3$; (b) $P=(0,9)$, the midpoint; (c) $Q=(6,11)$; (d) $D=(9,22)$; (e) $240$ units$^2$; (f) $F=(6,1)$.

Right-angled triangle completed to a rectangle

Right-angled triangle completed to a rectangle

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