Question

What is the orthocenter of the triangle formed by the intersection of the lines `x = 4` , `y = 1/2x + 7` , and`y = -x + 1`?

Answer 1

The Orthocenter, O is`=>O(0,5)`

Explanation:

The orthocenter is the point where all three altitudes of the triangle intersect. An altitude is a line which passes through a vertex of the triangle and is perpendicular to the opposite side. So we need to find lines that are perpendicular to the give equation and where at least 2 of them intersect:

A perpendicular line to any line, `y=mx+b` is give by:
`y_(per) = m_(per)x+ b_(per); m_(per)= -1/m` =====> (1)
First we need to get the coordinates of the vertices,
Let's solve equations:
Given `y_1=1/2x+7` and `x=4` simultaneously to get `y=11`
`y_2=-x+1` and `x=4` simultaneously to get `y=3`
`y=1/2x+7, y=-x+1; 1/2x+7=-x6+1; x=-4; y = 5`
Thus the vertices of the triangle are given by:
`A(-4,5), B(4,11)` and `C(4,-3)`

Now let find the perpendicular line to `y_1` that passes through point `C(4,-3)` from (1) `m_(per)= -1/(1/2)= -2`
`y_(per_1)= -2x+ b_(per)` let `x=4; y = -3 " and solve for " b_(per)`
`-3=-2(4)+ b_(per); b_(per)=5`
So what we have is `y_(per_1) = -2x+5`
and the line perpendicular to x=4 passing through A(-4,5) is `y=5`

Thus the Orthocenter is `y_(per_1) = -2x+5 = 5` Solve for `x`
`x= 0` `:. "the Orthocenter," =>O(0,5)`