Question

In a $△ABC,$ $A(-2,3),B(2,1)$ and $C(1,2)$. What is the foot of the altitude from the vertex $A$ of the triangle $ABC$?

• A. $(1,4)$
• B. $(-1,3)$
• C. $(-2,4)$
• D. $(-1,4)$

Answer 1

The correct option is D $(-1,4)$

Given : $A(-2,3),B(2,1)$ and $C(1,2)$

Slope of $BC=\frac{2-1}{1-2}=-1$

Equation of line $BC$ is $y-2=-1(x-1)$

$⟹x+y-3=0$

The foot of perpendicular $(h,k)$ from a point $(p,q)$ to a line $ax+by+c=0$ is given as

$\frac{h-p}{a}=\frac{k-q}{b}=\frac{-(ap+bq+c)}{a^2+b^2}$

The foot of perpendicular from $A(-2,3)$ to $x+y-3=0$ is

given by the following equation

$\frac{x+2}{1}=\frac{y-3}{1}=\frac{2}{1+1}$

$⟹x=-1,y=4$

Hence, foot of altitude $(x,y)=(-1,4)$