Problem
Counting segments, lines, and rays
Ten points lie on one line. A segment is named by its two endpoints, so AB and BA are the same. A ray is fixed by its start and direction, and there are only two directions. Count the segments and the rays.
Reasoning · Grade 3-1 Counting Figures Formed by Joining Dots
Your answer
Watch out Two rays per point suggests 20, but the two end points face only one way, leaving 18.
How to solve
Strategy Make a Systematic List — Both counts are 'how many' questions, so I list the possibilities in an orderly way. A segment cares only about which two points it joins (order does not matter). A ray cares about its starting point AND its direction, but on one straight line there are only two directions, so I count a ray by its start point and whether it heads left or right. I first try a tiny case (3 points) to find the counting rule, then scale up to 10 points.
1STEP 1
Shrink to an easier case: 3 points
With three points: 3 segments and 4 rays.
3 points: segments = 3, rays = 4
A segment is one connection between two points (order does not matter). A ray needs a start point and a direction, and on one line the only directions are left and right, so all the points reached the same way form a single ray.
4.G.A.1Solve An Easier Related Problem2STEP 2
Count the segments: unordered pairs
Segments are pairs: 9 + 8 + … + 1 = 45.
9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45
Pairing each point only with the points that come after it counts every pair exactly once, so no segment is double-counted.
3.OA.C.7Make A Systematic ListPairing each point only with the points that come after it in the row counts every segment exactly once.
Why?
A segment is named by its two endpoints and nothing else, so each pair of points names one segment and each segment has one pair.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
The points lie in a definite left-to-right order, so for any two of them exactly one comes first and the rule picks that one as the starter.
🧱Transitivity of orderIf the first is smaller than the second and the second smaller than the third, the first is smaller than the third, so pairwise comparisons chain into one full order.
3STEP 3
Count the rays: start point plus left/right direction
Eight middles face both ways, the ends one: 8 × 2 + 2 = 18.
8 × 2 + 2 × 1 = 16 + 2 = 18
Most points launch 2 rays (one each way), but the two end points launch only 1 ray each because there is nothing beyond them on that side. This is the same as 2 directions for every point except we lose one direction at each of the two ends: 2 × 10 - 2 = 18.
3.OA.A.1Make A Systematic List4STEP 4
Report the two counts in order
So 45 segments and 18 rays.
segments = 45, rays = 18
Checking the pattern of rays as the number of points grows (2, 4, 6, 8, ...) shows the rays equal 2 less than twice the number of points, and for 10 points that is 18.
3.OA.C.7Look For A PatternAnswer
45 segments and 18 rays
The small case checks out: 3 points give 3 segments and 4 rays, and the ray rule 2 x (points) - 2 gives 2 x 3 - 2 = 4. For 10 points the same rule gives 2 x 10 - 2 = 18, matching the direct count 8 x 2 + 2 = 18. The segment count 45 is the well-known choose-2 sum 1 + 2 + ... + 9 = 45. The two are different ideas: a segment needs two endpoints, but a ray on a line is fixed by its start and just one of two directions, so the through-point does not create extra rays.
Takeaway
A segment needs two endpoints, but a ray on a straight line only needs a starting point and one of two directions (left or right) - so the inside points make 2 rays each and the two end points make just 1 each, giving 18 rays!
- Shrink to an easier case: 3 points
- Count the segments: unordered pairs
- Count the rays: start point plus left/right direction
- Report the two counts in order