Geometry

Question Analysis

The main goal is to find the orthocenter of the triangle with vertices , , and . We need to 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: 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: It 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 , . This means  is a horizontal line.

Slope of : Given  and , , which is undefined. So,  is a vertical line.

Slope of : Given

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