The vertices of triangle ABC are A(1,3), B(7,7) and C(9,3) and I need to find the orthocenter.

Answer 1

Step 1:

The vertices of triangle ABC are ,   and .

Orthocenter:  Orthocenter of a triangle is the pont of intersection of all the altitudes of the triangle.

Slope of the line passing through the points  and  is .

Find the slope of the side  passing through   and .

.

Observe the figure:  and  are perpendicular.

If  and  are the slopes of two perpedicular lines then .

Slope of the altitude  is .

Find the line equation of the altitude .

Point-slope form of line equation is .

.

Step 2:

Find the slope of the side  passing through  and .

.

Observe the figure:  and  are perpendicular.

Find the line equation of the altitude .
Slope of the altitude  is .

Point-slope form of line equation is .

.

Solve (1) and (2) to find the orthocenter.

From (2): .

Substitute in (1).

Substitute  in 

.

Orthocenter is .

Solution:

Orthocenter is .