Geometry

Question Analysis

The main focus lies in determining the orthocenter of the triangle having vertices , , and . This involves calculating the slopes of the triangle's sides, finding slopes of the altitudes, formulating equations of the altitudes, and identifying 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:  helps find the equation of a line.

4. Orthocenter: It is the point where the altitudes of a triangle meet.

 

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 .

Slope of altitude from

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