Problem: ABC is a triangle: A=(0,0),B=(36,15) and both the coordinates of C are integers. What is the minimum area △ABC can have?
Answer Choices:
A. 21
B. 1
C. 23
D. 213
E. there is no minimum
Solution:
Since A=(0,0) and B=(36,15), we know that base AB of △ABC has length 3122+52=39. We must choose C so that the height of △ABC is minimum. The height is the distance to C=(x0,y0) from the line AB. This line is 5x−12y=0. In general, the distance from (x0,y0) to the line ax+by=c is
a2+b2∣ax0+by0−c∣.
So in this case the distance is ∣5x0−12y0∣/13. Since x0 and y0 are integers, the smallest this expression could be is 1/13. (0/13 is not possible, for then C would be on line AB and we would not have a triangle.) The value 1/13 is achieved, for instance, with C=(5,2) or (7,3). Thus the minimum area is (1/2)bh=(1/2)(39)(1/13)=3/2.
OR
The triangle with vertices (x1,y1),(x2,y2),(x3,y3) has area