Question

Find the equations of the altitudes of a $△ABC$ whose vertices are $A(1,4),B(-3,2)$ and $C(-5,-3)$.

Answer 1

$A(1,4),B(-3,2),C(-5,-3)$

$A(1,4),B(-3,2)$

Let CL be the altitude of AB

mid point $=(-1,3)$ slope of AB $=\frac{1}{2}\Rightarrow$ slope of CL $=-2$

$(y-3)=-2(x+1)$

$2x+y-1=0$..............A

$B(-3,2),C(-5,-3)$

Let AM be the altitude of BC

mid point $=(-4,-\frac{1}{2})$ slope of BC $=\frac{5}{2}\Rightarrow$ slope of AM $=-\frac{2}{5}$

$y+\frac{1}{2}=-\frac{2}{5}(x+4)$

$4x+10y+21=0$..............B

$C(-5,-3),A(1,4)$

Let BN be the altitude of CA

mid point $=(-2,\frac{1}{2})$ slope of CA $=\frac{7}{6}\Rightarrow$ slope of BN $=-\frac{6}{7}$

$y-\frac{1}{2}=-\frac{6}{7}(x+2)$

$12x+14y+17=0$..............C