# orthocenter

• Jan 8th 2009, 06:24 PM
misterriceman
orthocenter
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 (Rofl)
• Jan 9th 2009, 02:58 AM
Orthocentre
Hello misterriceman
Quote:

Originally Posted by misterriceman
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 (Rofl)

Suppose that the orthocentre, O, has coordinates $(a, b)$. Then we use the fact that the line joining the orthocentre to any vertex is perpendicular to the opposite side. So:

Gradient of OX = $\frac{b-51}{a-55}$

Gradient of YZ = $\frac{91}{39}$

So the product of these gradients is $-1$; i.e.

$\frac{91}{39}\times\frac{b-51}{a-55}= -1$ (1)

In a similar way, using OZ and XY, we get:

$\frac{47}{45}\times\frac{b-95}{a-49} = -1$ (2)

Now solve the simultaneous equations (1) and (2) to find $a$ and $b$. (Horrible numbers!) I make this:

$a = \frac{382}{3}$ and $b = 20$

I think you should check my working!