Question

Find the coordinates of the orthocenter of the triangle with the given vertices.
L (8,0) M (10,8) N (14,0)

Answer 1

The orthocentre is the intersection of the altitudes of a triangle.
You will need the equation of two of these and find their intersection

Make a sketch:
notice that LN lies on the x-axis, so the altitude from M is simply x = 10
that was too easy.

let's find the altitude to MN
slope of MN = (8-0)/(10-14) = -2
so slope of altitude is 1/2
and passing through (8,0)
0 = (1/2)(8) + b
b = -4
thus y = (1/2)x - 4
but x = 10
then y = (1/2)(10) - 4 = 1

the orthocentre is (10,1)

As an additional check, find the remaining altitude and verify by subbing in (10,1)