i need to find the orthocenter of a triangle
the cooridinates are:
X = 55, 51
Y= 10 , 4
Z = 49, 95
and if you can, can you show work
Thank You very much for your help
Hello misterriceman
Suppose that the orthocentre, O, has coordinates $\displaystyle (a, b)$. Then we use the fact that the line joining the orthocentre to any vertex is perpendicular to the opposite side. So:
Gradient of OX = $\displaystyle \frac{b-51}{a-55}$
Gradient of YZ = $\displaystyle \frac{91}{39}$
So the product of these gradients is $\displaystyle -1$; i.e.
$\displaystyle \frac{91}{39}\times\frac{b-51}{a-55}= -1$ (1)
In a similar way, using OZ and XY, we get:
$\displaystyle \frac{47}{45}\times\frac{b-95}{a-49} = -1$ (2)
Now solve the simultaneous equations (1) and (2) to find $\displaystyle a$ and $\displaystyle b$. (Horrible numbers!) I make this:
$\displaystyle a = \frac{382}{3}$ and $\displaystyle b = 20$
I think you should check my working!
Grandad