Question

Find the orthocecentre of the triangle with the vertices (-5,-7),(13,2),(-5,6)

Answer 1

A=(-5, -7) B=(13,2) C=(-5, 6)
Let the Orthocenter H be (x,y)

See that A and C have the same x coordinate. AC is parallel to y axis.

Let us shift the origin O(0,0) to A(-5,-7).
Hence new origin is O' = A itself.
We work now in the new coordinate system
(X= x+5, Y=y+7).

New coordinates
A (0,0),
B = (13--5, 2--7) = (18, 9)
C = (-5--5, 6--7) = (0, 13)

AC ⊥ X axis. So BE⊥ AC will be : Y = 9. So ordinate of H = 9.

Slope of AB: (9-0)/(18-0) = 1/2.
=> Slope of CD = -2.
{ Equation of CD ⊥ AB, }

passing through C: (Y - 13) / (X - 0) = - 2

Y + 2 X - 13 = 0
Substitute Y = 9,
2X = 13-9=4
X=2

Hnce H(2, 9) in the new coordinate system.

In the original coordinate system :
Orthocenter H = (2-5, 9-7)

H = (-3, 2)