Problem Solving

BTGmoderatorDC wrote:

In the given figure, if AS = 10 cm, SN = 5 cm and TN = 8 cm, which of the following could be the positive difference between the maximum and minimum value of AT, if all the sides shown in the figure are positive integers?

A) 16
B) 18
C) 20
D) 21
E) 23

Third-side rule:
The third side a triangle must be LESS THAN THE SUM OF and GREATER THAN THE DIFFERENCE OF the other two sides.
Here, all the sides must be INTEGER values.

In triangle ANS, AS=10 and NS=5.
Thus, third side AN must be an integer value less than 10+5 = 15 and greater than 10-5 = 5, yielding the following options for AN:
6, 7, 8, 9, 10, 11, 12, 13, 14

In triangle ANT, TN=8.
If AN=8, then third side AT must be an integer value less than 8+8=16 and greater than 8-8=0, making it possible that AT=1.
Since AT must be a positive value, the minimum possible value for AT=1.
If AN=14, then third side AT must be an integer value less than 14+8=22 and greater than 14-8=6, making it possible that AT=21.
Since any other value for AN will yield a lower upper limit than the blue equation above, it is not possible that AN>21.
Implication:
The greatest possible value for AN=21.
Thus:
Maximum - minimum = 21-1 = 20.

The correct answer is C.