Geometry

Question Analysis

The central task here is to determine the orthocenter of the triangle with vertices , , and . To achieve this, we calculate the slopes of the triangle's sides, find the slopes of the altitudes (using the property of perpendicular lines), formulate the equations of the altitudes, and then identify their intersection point.

 

Key Concept Explanation

1. Slope formula: For two points  and , the slope .

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

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

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

 

Step - by - Step Solution

1. Calculate slopes of triangle sides:

Slope of : Given  and , .

Slope of : Given  and , .

Slope of : Given  and , .

2. Determine slopes of altitudes:

Slope of altitude from  to : Since , the slope of this altitude

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