Geometry

Question Analysis

The primary focus is to determine the orthocenter of the triangle formed by , , and . We need to calculate the slopes of the triangle’s sides, derive the slopes of the altitudes, formulate 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 where all the altitudes of a triangle intersect.

 

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 .

Slope of altitude from