Geometry

Question Analysis

The main focus lies in calculating the orthocenter of the triangle with vertices , , and . This involves determining the slopes of the triangle's sides, the slopes of the altitudes, formulating the equations of the altitudes, and finding their intersection point.

Key Concept Explanation

1. Slope formula: Given two points  and , .

2. Perpendicular lines: If two lines with slopes  and  are perpendicular, .

3. Point - slope form: The equation of a line passing through  with slope  is .

4. Orthocenter: The orthocenter is the point of intersection of the altitudes of a triangle.

Step - by - Step Solution

1. Find slopes of triangle sides:

Slope of : Given  and , .

Slope of : Given  and , . So,  is a horizontal line.

Slope of : Given  and ,