Question

Triangle XYZ has vertices X(0,0), Y(-4,0), and Z(0,6). Find the coordinates of the indicated point.

A) the centroid of triangle XYZ

B) the orthocenter of triangle XYZ

Answer 1

Can anybody explain how to solve this?

Answer 2

Centroid of a triangle is\[(\frac{ a+b+c }{3 },\frac{ x+y+z }{ 3 })\]
where the co-ordinates are (a,x) (b,y) (c,z).

Answer 3

(0+ (-4) + 0)/ 3 , (0+0+6) /3 like this?

Answer 4

As it is a right angled triangle, the orthocentre is the vertex X itself.

Answer 5

Yeah, correct.

Answer 6

so (-4/3, 2) is the centroid?

Answer 7

Yes, and did you understand the orthocentre part?

Answer 8

no not really:( sorry...

Answer 9

What is the definition of orthocentre?

Answer 10

the lines that contain altitudes and intersect at a point are called the orthocenter of the triangle

Answer 11

Altitude is a line drawn from vertex to opposite side perpendicularly.
Here, X=90 degrees. So, XY and XZ are altitudes..
Getting it?

Answer 12

I think so. so the coordinates off the centroid are (-4/3, 2) right? do I have to graph it?

Answer 13

Your centroid is correct. I don't think there is a need to graph it.

Answer 14

and the orthocenter? the coordinates ? Sorry I don't understand

Answer 15

What are the altitudes drawn from Y and Z ?

Answer 16

XY and XZ ?

Answer 17

Yes, So their point of intersection is the orthocentre...

Answer 18

Thank you so much!!!

Answer 19

And you should also know the general method of finding the orthocentre. Do you?

Answer 20

no :(

Answer 21

Okay, read this..
1. If you can find any two altitudes, it is enough for getting the orthocentre. The third one passes through the orthocentre.
2. So, find two altitudes. Then their point of intersection.
3. You can get the equation of altitude by taking a side,find the slope of line perpendicular to it, then use the slope-intercept form using the opposite vertex.