Slanted segment on a grid – length of a segment in two steps
Each task has two steps for one segment AB: its length, and then a comparison with a given number. After every answer you see the right method next to a typical mistake, and a picture that shows why.
What next?
How do you play?
Two points, A and B, are marked on a grid, with the segment AB between them. The side of one grid cell is 1 long. In step 1 you choose the length of AB from four answers. In step 2 you compare AB with a given number: you choose whether AB is longer, shorter or exactly that long. After every answer you see the right method next to the typical mistake behind one of the other answers, and a picture: a square built on the segment AB, or a circle around point A. There are 5 tasks. At the end you will see where to find more practice. By default there is no time limit; the timed mode is optional and only counts the time while a question is waiting for your answer.
How do you find the length of a segment on a grid?
Count how many cells there are between the points across and how many up. These two numbers are the legs of a right triangle in which AB is the hypotenuse. By the Pythagorean theorem: AB² = (cells across)² + (cells up)². For A = (1, 2) and B = (7, 10) there are 6 cells across and 8 up, so AB² = 36 + 64 = 100 and AB = 10.
When the result is not a square of a whole number, leave the square root: for 5 cells across and 4 up, AB² = 25 + 16 = 41, so AB = √41. This is an irrational number, and its value lies between 6 and 7, because 6² = 36 and 7² = 49.
How do you count cells when the coordinates are negative?
Subtract the coordinates together with their signs and put the second number in brackets so that you do not lose the minus. For x = 4 and x = −2: 4 − (−2) = 6, so 6 cells across. When the points lie on opposite sides of an axis, the distance is the sum of the distances from the axis. Do not subtract the absolute values of the coordinates: |4| − |−2| = 2, and that is not the distance between the points.
How do you compare a square root with a number?
Both numbers are positive, so compare their squares. Is √90 greater than 10? The square of √90 is 90 and the square of 10 is 100. Since 90 < 100, √90 < 10. To be sure, estimate: 9² = 81 and 10² = 100, and 90 lies between them, so √90 is between 9 and 10.
What is the most common mistake?
Adding the cells across and up, which gives the path along the grid lines instead of the slanted segment: for 6 and 8 this gives 14, but the real length of the segment is 10. The slanted segment is always shorter than the path along the grid lines. Another common mistake is comparing the square of the length with the number itself: 90 with 10 instead of 90 with 100. Then the segment seems “longer than 10” although it is shorter.
How does this help in the eighth-grade exam?
In the Polish eighth-grade maths exam in 2026, question 12 (true–false) required finding the length of a segment with endpoints at grid points of a coordinate system and applying the Pythagorean theorem. 45% of pupils solved it correctly. 63% judged the first statement correctly (it needed two lengths equal to 5), and only 36% judged the second one, which needed a comparison of √90 with the number 10.
Source: CKE report 2026, pp. 22–23 (in Polish). Curriculum: coordinate system (grades 7–8, X.5) and the Pythagorean theorem (Polish core curriculum, Dz.U. 2024 poz. 996).
Mini glossary: English – Polish
- segment – odcinek
- length – długość
- Pythagorean theorem – twierdzenie Pitagorasa
- hypotenuse – przeciwprostokątna
- square root – pierwiastek kwadratowy
- coordinate system – układ współrzędnych
The game runs in your browser. There is a time limit only in the timed mode, there are no negative points and we save nothing.