Question

The midpoints $D,E,F$ of sides $BC,CA$ and $AB$ of a $△ABC$ are $(2,-1),(1,3)$ and $(-1,-2)$ respectively. The slope of the altitude through $B$ is :

• A. $3$
• B. $-2$
• C. $1/3$
• D. $5/2$

Answer 1

The correct option is B $-2$

REF.Image

$\frac{\alpha +r}{2}=1$ ___ (1)

$\frac{\alpha +\lambda }{2}=-1$ ___ (2)

$\frac{\lambda +r}{2}=2$ ___ (3)

$\frac{\beta +\delta }{2}=3$ ___ (4)

$\frac{\beta +\mu }{2}=-2$ ___ (5)

$\frac{\mu +\delta }{2}=-1$ ___ (6)

Subtract (2) and (3),

$\alpha -r=-6$

$\alpha +r=2$

______________

$2\alpha =-4$

$\alpha =-2\Rightarrow r=2$

Subtract (5) and (6)

$\beta -\delta =-2$

$\beta +\delta =6$

_____________

$2\beta =4$

$\beta =2$

$\Rightarrow \delta =4$

$\Rightarrow M_{AC}=1/2$

$M_{AC}=\frac{4-2}{2+2}=2/4=1/2$

$M_{BD}$ = slope of altitude from B

$=\frac{-1}{M_{AC}}=-2$

$[\because m_1{m}_2=-1$ if two lines are

$\perp$.]