Triangle ABC has vertices A(8,2), B(0,6), and C(-3,2)

Triangle ABC has vertices A(8,2), B(0,6), and C(-3,2). Point C can be moved along a certain line, with points A and B remaining stationary, and the area of ABC will not change. What is the slope of that line?

1. -1/2
2. -3/4
3. 0
4. 4/3
5. 2

Answer

The correct answer is A.

Because points A and B are stationary, the side AB of ΔABC will not change in length or direction. The altitude of ΔABC with respect to side AB is the segment having one endpoint at point C that is perpendicular to AB.

Because the length AB must remain constant, the altitude from C of any new triangle whose area is equal to that of the original ΔABC must have the same length. In order that all altitudes from the moving point C have equal lengths, C must lie on the line which parallel to AB.

Therefore, the slope is same as the slope of AB.

Slope of AB = (6-2)/(0-8) = -1/2