Geometry

Question Analysis

The key focus is to determine the orthocenter of the triangle having vertices , , and . We calculate slopes of the triangle's sides, then slopes of the altitudes. Afterward, we form equations of the altitudes and find their intersection.

 

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:  is used to find the equation of a line.

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

 

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