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)

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- Jan 8th 2009, 06:24 PMmisterricemanorthocenter
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 AMGrandadOrthocentre
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

- Jan 10th 2009, 07:57 AMmisterriceman
thank u very muchhh

i love u <3333333