Question

Need urgent help:
In Triangle ABC,
A (-9, -2)
B (-3, 4)
C (3, -2)
Find the Perpendicular slopes of AB, CB, and AC, then find the coordinates of the orthocenter.

I have found the slopes already and got
Perpendicular slope of AB is -1
Perpendicular slope of CB is Zero
Perpendicular slope of AC is Undefined

I have no idea how to graph this and find the orthocenter (where all the perpendicular slopes meet). Please help!

Answer 1

Slope 1: y2 - y1 / x2 - x1 (AB)
= 4 - -2 / -3 - -9
= 4 + 2 / -3 + 9
= 6 / 6
= 1

Slope 2: y2 - y1 / x2 - x1 (BC)
= -2 - 4 / 3 - -3
= -6 / 3 + 3
= -6 / 6
= -1

Slope 3: y2 - y1 / x2 - x1 (AC)
= -2 - -2 / 3 - -9
= -2 + 2 / 3 + 9
= 0 / 12
= 0
Slope perpendicular to AB: -1/1
Slope perpendicular to BC: 1/1
Slope perpendicular to AC: undefined

Answer 2

Thanks for answering that one, it was making my eyes hurt :)

Answer 3

Oh, oops small mistake on slope of CB, thanks. But how do I find the orthocenter?

Answer 4

The orthocenter has an interesting property of being the point where the altitudes of the sides of the triange intersect. Now, if you could define a line from each vertex of the triangle that was perpendicular to the opposite side of the respective vertex algebraically (and the lines be in y=mx+b form), you could set them equal to each other and solve for the x-value where the orthocenter is. Then, you could re-apply that x-value into one of your equations to find the respective y.

Answer 5

It is kind of like making a system of linear equations for each altitude line and finding the solution.

Answer 6

If you already plotted the points, how could you use the perpendicular slopes to find the orthocenter? Do you HAVE to use an equation?

Answer 7

For example I have A (-9, -2) plotted on a graph. The perpendicular slope of AB is -1, so do I go 1 down, 1 to the right from A? How would I do it?

Answer 8

That could work too, although you still use the opposite side's perpendicular slope. (The line going through the vertex is perpendicular to the line not included with the vertex).
IE, you use the slope of the perpendicular to BC for vertex A

Answer 9

So how would I find the orthocenter with that?

Answer 10

oh flutter... this is confusing.

Answer 11

Graph looks something like this?

Answer 12

I drew lines from each vertex with the same slope as the opposite sides' perpendicular and they appear to intersect at (-3,4)