Question

What are the coordinates of the orthocenter of triangle ABC with vertices at A(1, 2), B(1, 6), and C(5, 6)?

Answer 1

The orthocenter of a triangle is the intersection of the triangle's three altitudes. This is shown below:

To find the orthocenter of the triangle, we need to find the equation of the lines of the triangle and find the lines perpendicular to each of them. We will only need to do this for two sides.

The equation of a line is given to be:

`y=mx+b`

where m is the slope and b is the intercept.

The slope is calculated to be:

`m=(y_2-y_1)/(x_2-x_1)`

Two lines are perpendicular if the product of their slopes is equal to -1, such that:

`\begin{gathered} m_1\cdot m_2=-1 \\ or \\ m_1=-(1)/(m_2) \end{gathered}`

Therefore, the slopes of perpendicular lines are inverse reciprocals of each other.

Equation of Line AB:

`\begin{gathered} A\to(x_1,y_1)=(1,2) \\ B\to(x_2,y_2)=(1,6) \end{gathered}`

The slope of the line is:

`m=(6-2)/(1-1)=(4)/(0)`

This means that the line is a vertical line. Therefore, the equation of the line is