Geometry

Question Analysis

The primary goal is to determine the orthocenter of the triangle formed by points , , and . We achieve this by calculating the slopes of the triangle's sides, then finding the slopes of the altitudes. After that, we generate the equations of the altitudes and solve for their intersection point.

 

Key Concept Explanation

1. Slope formula: Given 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 is the point where the altitudes of a triangle intersect.

 

Step - by - Step Solution

1. Calculate slopes of the sides:

Slope of : Given  and , .

Slope of : Given  and , .

Slope of : Given  and , .

2. Determine slopes of the altitudes:

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

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