Question

The coordinates of the feet of the perpendiculars from the vertices of a triangle on the opposite sides are $(20,25),(8,16)$ and $(8,9)$. The coordinates of a vertex of the triangle are

• A. (5, 10)
• B. (50, -5)
• C. (15, 30)
• D. (10, 15)

Answer 1

Answer: (A), (B), (C)

The correct options are
A

(5, 10)

B

(50, -5)

C

(15, 30)

We use the fact that the orthocentre $O$ of the triangle $ABC$ is the incentre of the pedal triangle $DEF$.
Let $(h,k)$ be the coordinates of $O$.
Now $ED=\sqrt{(20-8{)}^2+(25-16{)}^2}=15$

$FD=20$ and $EF=7$
so that $h=\frac{7\times 20+20\times 8+15\times 8}{7+20+15}=10$
and $k=\frac{7\times 25+20\times 16+15\times 9}{7+20+15}=15$
thus, the coordinates of $O$ are $(10,15)$
Since $AC$ is perpendicular to $OE$, equation of $AC$ is
$y-16=-\frac{10-8}{15-6}(x-8)\Rightarrow y-2x=0\left(1\right)$
Similarly equation of $AB$ is
$y-9=-\frac{10-8}{15-9}(x-8)\Rightarrow 3y+x-35=0\left(2\right)$
and equation of $BC$ is
$y-25=-\frac{20-10}{25-15}(x-20)\Rightarrow y+x-45=0\left(3\right)$
Solving (1) and (2) we get $A(5,10)$
From (2) and (3) we get $B(50,-5)$ and from (3) and (1) we get $C(15,30)$