23. Five points lie on a straight line. Alex finds the distances between every pair of points. He obtains, in increasing order, $2,5,6,8,9, k, 15,17,20$ and 22 . What is the value of $k$ ?
A 14
B 13
C 12
D 11
E 10
23. A Let the five points be $P, Q, R, S$ and $T$ with individual distances between them of $w, x, y$ and $z$ as shown.
The maximum distance between any two points is 22 so $P T=22$. The next largest distance is 20 and, since no distance is 1 , this is either $P S$ or $Q T$. Assume $Q T=20$ so that $P Q=w=2$. The next largest distance is 17 and, since no distance is 3 this cannot be $Q S$ or $R T$ and so is $P S$. Hence $S T=z=5$ and $Q S=15$. The remaining distances are $6,8,9$ and $k$ which represent the lengths of $P R, Q R, R S$ and $R T$ in some order with $Q R+R S=15$. Since $k>9$, the only possible pair of distances adding to 15 is 6 and 9 and so $Q R=x=6$ and $R S=y=9$ (since $Q R=9$ and $R S=6$ would mean two distances are 11) leaving $P R=8$ and $R T=k=14$. Hence the value of $k$ is 14 with the distances between the points taking the values shown below.
(Note: The same distances would result but in the reverse order if we assumed $P S=20$.)