help me on this question. i don't get what is orthocentre.?

Question

Triangle DEF has vertices D ( -4, 6) E (0,2) and F ( 8,6)
determine the coordinates of the ORTHOCENTRE of triangle ABC

Answer 1

The plan:

find the slope of line DE using points D & E.
multiply it by -1 and take the reciprocal to get the slope of the first perpendicular line to DE then use coordinates of point F with the new slope to find an equation for this new line.
repete the process using points F&E or D&F to get a slope, multiply by -1 and take its reciprocal then use it with the third point to get another linear equation then solve the first & second linear equations simultaniously to get the intersection points (coordinates of orthocenter).

Long process but here we go:

(6-2)(-4-0)=4/-4= -1 slope of DE
-1*-1=1 slope of one of the ortho lines, since it goes through
F then F is a point on the line. so line equation is 1=(y-8)/(x-6)

or y= x+2

do the same for EF then using point D to get

slope EF= (2-6)/(0-8)= 1/2 so new slope is -1(1/(1/2))= -2
then -2=(y-6)/ (x+4)

y= -2x-2 = x+2 x=-4/3 & y=2/3

Answer 2

Hi,

this is the same orthocenter answer I gave on your other 3 part question. I assume you wanted triangle DEF, not triangle ABC.

1) The orthocenter of a triangle is the point where the three altitudes meet. We will find the intersection of 2 altitudes to find the orthocenter.

Side DE has a slope of (6 - 2)/(-4 - 0) or -1. A perpendicular line would have a slope that is the negative reciprocal of -1, which is a slope of 1. The altitude to side DE would have to go through point F (8,6). Its formula in point-slope form is
y - 6 = 1(x - 8) which simplifies to y = x - 2.

Side EF has a slope of (6 - 2)/(8 - 0) or ½. A perpendicular line would have a slope that is the negative reciprocal of ½, which is a slope of -2. The altitude to side EF would have to go through point D (-4,6). Its formula in point-slope form is
y - 6 = -2(x - (-4)) which simplifies to y = -2x - 2.

Two of the altitudes have the equations:

y = x - 2
y = 2x - 2

Since they both equal "y", the 2 right sides can be set equal to each other.

x - 2 = 2x - 2
x = 0

Substituting x = 0 into either equation will find the value of y at their intersection. y = 0 - 2 so y = 2.

The point of intersection for the altitudes, which is the location of the orthocenter, is (0,2).

I hope that helps again!! :-)

Answer 3

the orthocenter is the point where if you drew a line from each corner to the mid point of the opposite side for all 3 sides

the point where they meet would be called the 'orthocenter'