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 can be

• 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 $△ABC$ is the incentre of the pedal triangle $△DEF.$

Let $(h,k)$ be the coordinates of $O$
$ED=\sqrt{{(20-8)}^2+{(25-16)}^2}=15$
$FD=20$ and $EF=7$
So that $h=\frac{7\times 20+20\times 8+15\times 8}{7+20+15}=10$ and $k=\frac{7\times 25+20\times 16+15\times }{7+20+15}=15$
Thus the coordinates of $O$ are $(10,15)$.

Since $AC$ is perpendicular to $OE$, equation of $AC$ is
$y-16=\frac{10-8}{15-16}(x-8)$
$\Rightarrow y-2x=0$ .......( 1)
Similarly equation of $AB$ is
$y-9=-\frac{10-8}{15-9}(x-8)$
$\Rightarrow 3x+x-35=0$ .....(2)
and equation of BC is
$y-25=-\frac{20-10}{25-15}(x-20)$
$\Rightarrow y+x-45=0$ .......(3)
Solving (1) and (2), we get

$A\equiv (5,10)$
From (2) and (3), we get $B\equiv (50,-5)$ and from (3) and (1), we get $C\equiv (15,30)$.