Geometry

Question Analysis

The main focus is to find the orthocenter of the triangle with vertices , , and . We do this by calculating slopes of the triangle's sides, determining slopes of the altitudes, formulating equations of the altitudes, and finding their intersection point.

 

Key Concept Explanation

1. Slope formula: For 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’s the intersection of 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

Click "Show Answer" to reveal the answer and analysis