Geometry

Question Analysis

The central task is to find the orthocenter of the triangle formed by points , , and . We must calculate the slopes of the triangle's sides, determine the slopes of the altitudes, write the equations of the altitudes, and then find their intersection point.

 

Key Concept Explanation

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

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: It is the point of intersection of all the altitudes of a triangle.

 

Step - by - Step Solution

1. Find 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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