Five points lie on the same line. When we list the ten distances between any two of these points, from smallest to largest, we get $2,4,5,7,8, k, 13,15,17,19$. What is the value of $k$?
Solution 1: - This solution is a bit difficult to write because it is based on "trial and error." We start by drawing a number line and placing the points 0 and 19. Since the first distance is 2, we mark our first three points:

![](https://cdn.mathpix.com/cropped/2024_05_01_2364a1ff751c3bd5ab7eg-6.jpg?height=112&width=962&top_left_y=655&top_left_x=653)

Since we need a distance of 7, we place the point 7 on the number line. This gives us distances that are not incompatible with the problem:

![](https://cdn.mathpix.com/cropped/2024_05_01_2364a1ff751c3bd5ab7eg-6.jpg?height=103&width=1011&top_left_y=1045&top_left_x=631)

The distances between these 4 points are: 2, 7, 19, 5, 17, and 12. Finally, placing the point 15 on the number line, we get the following:

![](https://cdn.mathpix.com/cropped/2024_05_01_2364a1ff751c3bd5ab7eg-6.jpg?height=126&width=1020&top_left_y=1425&top_left_x=629)

With these points, the distances are: \(2, 7, 15, 19, 5, 13, 17, 8, 12, 4\), which are compatible with the data in the problem. Therefore, \(k=12\).

Note that we also have another distribution of the numbers, namely:

![](https://cdn.mathpix.com/cropped/2024_05_01_2364a1ff751c3bd5ab7eg-6.jpg?height=123&width=1033&top_left_y=1903&top_left_x=617)

In this distribution, we also get \(k=12\).

Solution 2: Since the largest distance is 19, we can assume that one point is 0 and the other is 19.

If \(a\) is another point, then in the list of distances, we have the numbers: \(a-0 = a\) and \(19-a\). Indeed, in the list, we have the pairs 2 and 17, so we can assume that the number 2 is another point on the number line.

Similarly, since 4 and 15 are in the list of distances, 4 or 15 is another point on the number line. However, 4 cannot be one of the points because the distance 2 did not appear twice. Therefore, 15 is another point on the number line.

Finally, the fifth point must be 5 units away from one of the points and 7 units away from another, so the missing point is the point 7, and the unknown distance is \(k=19-7=12\).