Question

If in a non-right-angled triangle ABC, tan A and tan B are rational and the vertices A and B are also rational points, then the correct statements is/are 

(A)  tan C must be rational

(B) C must be a rational point

(C) tan C may be irrational

(D) C may be an irrational point

Answer 1

Correct option  (A) , (B)

tan A + tan B + tan C = tan A tan B tan C

⇒ tan C = tan A + tan B/tan A tan B -1

⇒ tan C must be rational

Since the slope of line AB is rational and tan A is also rational, therefore slope of line AC is also rational. Similarly, slope of line BC is also rational.

⇒ C must be a rational point.