id
stringlengths 46
46
| setting_key
stringclasses 1
value | setting_name
stringclasses 0
values | messages
listlengths 1
1
| ground_truth
stringclasses 15
values | dataset
stringclasses 1
value |
|---|---|---|---|---|---|
geometry_perpendicular_intersection_train_0000
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 4, from vertex 4 to vertex 8, from vertex 10 to vertex 4, from vertex 6 to vertex 3, from vertex 5 to vertex 8, from vertex 7 to vertex 9, from vertex 9 to vertex 2, from vertex 2 to vertex 6, from vertex 10 to vertex 5, from vertex 4 to vertex 6, from vertex 1 to vertex 3, and from vertex 4 to vertex 9. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
7
|
math
|
geometry_perpendicular_intersection_train_0001
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 9-gon, with vertices numbered 1 through 9 in counterclockwise order. Draw the diagonal from vertex 8 to vertex 1, from vertex 5 to vertex 8, from vertex 8 to vertex 3, from vertex 9 to vertex 5, from vertex 2 to vertex 5, from vertex 9 to vertex 2, from vertex 2 to vertex 8, from vertex 4 to vertex 7, from vertex 1 to vertex 4, from vertex 5 to vertex 7, from vertex 8 to vertex 6, and from vertex 6 to vertex 1. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
0
|
math
|
geometry_perpendicular_intersection_train_0002
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 9-gon, with vertices numbered 1 through 9 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 8, from vertex 9 to vertex 5, from vertex 4 to vertex 6, from vertex 7 to vertex 4, from vertex 3 to vertex 9, from vertex 2 to vertex 5, from vertex 6 to vertex 2, from vertex 7 to vertex 5, from vertex 8 to vertex 4, from vertex 1 to vertex 6, and from vertex 9 to vertex 4. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
0
|
math
|
geometry_perpendicular_intersection_train_0003
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 11 to vertex 4, from vertex 3 to vertex 12, from vertex 8 to vertex 1, from vertex 1 to vertex 4, from vertex 7 to vertex 2, from vertex 12 to vertex 10, from vertex 3 to vertex 10, and from vertex 5 to vertex 10. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
6
|
math
|
geometry_perpendicular_intersection_train_0004
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 11-gon, with vertices numbered 1 through 11 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 4, from vertex 3 to vertex 5, from vertex 10 to vertex 8, from vertex 5 to vertex 10, from vertex 5 to vertex 2, from vertex 3 to vertex 8, from vertex 8 to vertex 11, from vertex 3 to vertex 11, from vertex 4 to vertex 7, and from vertex 9 to vertex 1. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
0
|
math
|
geometry_perpendicular_intersection_train_0005
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 8, from vertex 2 to vertex 7, from vertex 6 to vertex 4, from vertex 3 to vertex 8, from vertex 5 to vertex 2, from vertex 3 to vertex 5, from vertex 4 to vertex 7, from vertex 5 to vertex 8, from vertex 6 to vertex 3, and from vertex 5 to vertex 7. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
6
|
math
|
geometry_perpendicular_intersection_train_0006
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 1, from vertex 3 to vertex 1, from vertex 4 to vertex 6, from vertex 6 to vertex 1, from vertex 3 to vertex 8, from vertex 1 to vertex 7, from vertex 5 to vertex 7, and from vertex 7 to vertex 3. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
5
|
math
|
geometry_perpendicular_intersection_train_0007
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 10, from vertex 5 to vertex 7, from vertex 6 to vertex 2, from vertex 7 to vertex 1, from vertex 10 to vertex 8, from vertex 7 to vertex 9, from vertex 8 to vertex 6, from vertex 9 to vertex 4, from vertex 6 to vertex 9, from vertex 5 to vertex 2, and from vertex 1 to vertex 5. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
4
|
math
|
geometry_perpendicular_intersection_train_0008
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 7-gon, with vertices numbered 1 through 7 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 2, from vertex 7 to vertex 3, from vertex 2 to vertex 7, from vertex 1 to vertex 5, from vertex 3 to vertex 5, from vertex 4 to vertex 1, from vertex 1 to vertex 6, and from vertex 6 to vertex 4. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
0
|
math
|
geometry_perpendicular_intersection_train_0009
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 8 to vertex 5, from vertex 1 to vertex 3, from vertex 3 to vertex 5, from vertex 3 to vertex 8, from vertex 5 to vertex 1, from vertex 3 to vertex 7, from vertex 5 to vertex 2, from vertex 6 to vertex 8, from vertex 4 to vertex 2, from vertex 7 to vertex 5, and from vertex 4 to vertex 6. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
9
|
math
|
geometry_perpendicular_intersection_train_0010
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 11 to vertex 4, from vertex 7 to vertex 12, from vertex 4 to vertex 10, from vertex 7 to vertex 4, from vertex 3 to vertex 7, from vertex 5 to vertex 12, and from vertex 7 to vertex 1. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
2
|
math
|
geometry_perpendicular_intersection_train_0011
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 10 to vertex 3, from vertex 2 to vertex 7, from vertex 2 to vertex 8, from vertex 7 to vertex 3, from vertex 6 to vertex 3, from vertex 4 to vertex 9, from vertex 6 to vertex 2, from vertex 6 to vertex 9, from vertex 10 to vertex 6, from vertex 7 to vertex 5, from vertex 1 to vertex 9, and from vertex 1 to vertex 8. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
5
|
math
|
geometry_perpendicular_intersection_train_0012
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 11, from vertex 2 to vertex 4, from vertex 5 to vertex 1, from vertex 1 to vertex 8, from vertex 11 to vertex 4, from vertex 4 to vertex 1, from vertex 1 to vertex 10, from vertex 6 to vertex 9, from vertex 9 to vertex 1, and from vertex 2 to vertex 6. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
4
|
math
|
geometry_perpendicular_intersection_train_0013
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 7 to vertex 2, from vertex 5 to vertex 1, from vertex 5 to vertex 8, from vertex 7 to vertex 3, from vertex 7 to vertex 1, from vertex 3 to vertex 8, from vertex 4 to vertex 8, from vertex 1 to vertex 6, from vertex 6 to vertex 4, from vertex 9 to vertex 1, from vertex 2 to vertex 6, from vertex 5 to vertex 2, and from vertex 7 to vertex 9. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
6
|
math
|
geometry_perpendicular_intersection_train_0014
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 9 to vertex 3, from vertex 2 to vertex 7, from vertex 4 to vertex 1, from vertex 4 to vertex 10, from vertex 6 to vertex 2, from vertex 4 to vertex 8, from vertex 8 to vertex 10, from vertex 5 to vertex 3, from vertex 9 to vertex 4, from vertex 1 to vertex 5, from vertex 3 to vertex 6, and from vertex 6 to vertex 9. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
5
|
math
|
geometry_perpendicular_intersection_train_0015
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 8, from vertex 6 to vertex 2, from vertex 4 to vertex 1, from vertex 8 to vertex 4, from vertex 4 to vertex 2, from vertex 3 to vertex 6, from vertex 6 to vertex 1, from vertex 2 to vertex 7, from vertex 3 to vertex 8, from vertex 2 to vertex 5, from vertex 7 to vertex 5, and from vertex 5 to vertex 8. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
8
|
math
|
geometry_perpendicular_intersection_train_0016
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 7, from vertex 4 to vertex 1, from vertex 1 to vertex 6, from vertex 6 to vertex 8, from vertex 8 to vertex 2, from vertex 5 to vertex 1, from vertex 3 to vertex 6, from vertex 4 to vertex 2, and from vertex 3 to vertex 5. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
5
|
math
|
geometry_perpendicular_intersection_train_0017
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 5, from vertex 5 to vertex 8, from vertex 7 to vertex 2, from vertex 5 to vertex 1, from vertex 4 to vertex 1, from vertex 4 to vertex 2, and from vertex 3 to vertex 7. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
4
|
math
|
geometry_perpendicular_intersection_train_0018
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 9, from vertex 4 to vertex 6, from vertex 1 to vertex 3, from vertex 7 to vertex 3, from vertex 2 to vertex 9, from vertex 5 to vertex 10, and from vertex 3 to vertex 10. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
4
|
math
|
geometry_perpendicular_intersection_train_0019
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 7 to vertex 1, from vertex 5 to vertex 1, from vertex 3 to vertex 5, from vertex 6 to vertex 4, from vertex 3 to vertex 6, from vertex 7 to vertex 2, from vertex 4 to vertex 8, and from vertex 3 to vertex 1. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
5
|
math
|
geometry_perpendicular_intersection_train_0020
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 3, from vertex 3 to vertex 5, from vertex 4 to vertex 6, from vertex 8 to vertex 3, from vertex 5 to vertex 8, from vertex 7 to vertex 3, from vertex 4 to vertex 1, from vertex 4 to vertex 7, and from vertex 2 to vertex 6. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
2
|
math
|
geometry_perpendicular_intersection_train_0021
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 10 to vertex 8, from vertex 9 to vertex 5, from vertex 10 to vertex 5, from vertex 2 to vertex 6, from vertex 1 to vertex 4, from vertex 7 to vertex 4, from vertex 7 to vertex 10, and from vertex 8 to vertex 4. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
1
|
math
|
geometry_perpendicular_intersection_train_0022
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 8 to vertex 1, from vertex 4 to vertex 6, from vertex 8 to vertex 5, from vertex 8 to vertex 3, from vertex 10 to vertex 5, from vertex 9 to vertex 7, from vertex 2 to vertex 5, from vertex 9 to vertex 6, from vertex 2 to vertex 7, and from vertex 6 to vertex 8. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
5
|
math
|
geometry_perpendicular_intersection_train_0023
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 7 to vertex 4, from vertex 5 to vertex 2, from vertex 4 to vertex 10, from vertex 6 to vertex 2, from vertex 10 to vertex 8, from vertex 3 to vertex 1, from vertex 2 to vertex 10, from vertex 8 to vertex 5, from vertex 9 to vertex 7, from vertex 9 to vertex 5, from vertex 1 to vertex 4, from vertex 6 to vertex 9, and from vertex 7 to vertex 2. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
7
|
math
|
geometry_perpendicular_intersection_train_0024
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 6, from vertex 7 to vertex 3, from vertex 3 to vertex 12, from vertex 10 to vertex 3, from vertex 9 to vertex 7, from vertex 11 to vertex 2, from vertex 9 to vertex 6, from vertex 6 to vertex 10, from vertex 12 to vertex 2, from vertex 4 to vertex 1, from vertex 12 to vertex 10, and from vertex 4 to vertex 7. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
6
|
math
|
geometry_perpendicular_intersection_train_0025
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 8 to vertex 6, from vertex 6 to vertex 4, from vertex 2 to vertex 8, from vertex 1 to vertex 6, from vertex 2 to vertex 7, from vertex 7 to vertex 10, from vertex 1 to vertex 3, from vertex 9 to vertex 7, from vertex 7 to vertex 4, from vertex 5 to vertex 2, and from vertex 9 to vertex 2. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
4
|
math
|
geometry_perpendicular_intersection_train_0026
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 6, from vertex 8 to vertex 3, from vertex 7 to vertex 1, from vertex 5 to vertex 8, from vertex 7 to vertex 3, from vertex 4 to vertex 1, from vertex 8 to vertex 2, from vertex 4 to vertex 7, from vertex 2 to vertex 4, from vertex 5 to vertex 3, from vertex 3 to vertex 1, from vertex 4 to vertex 6, and from vertex 6 to vertex 3. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
9
|
math
|
geometry_perpendicular_intersection_train_0027
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 5, from vertex 6 to vertex 3, from vertex 4 to vertex 6, from vertex 1 to vertex 3, from vertex 1 to vertex 4, from vertex 6 to vertex 2, and from vertex 4 to vertex 2. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
4
|
math
|
geometry_perpendicular_intersection_train_0028
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 8 to vertex 6, from vertex 3 to vertex 7, from vertex 5 to vertex 8, from vertex 4 to vertex 7, from vertex 8 to vertex 1, from vertex 7 to vertex 5, from vertex 3 to vertex 1, from vertex 10 to vertex 6, and from vertex 9 to vertex 2. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
4
|
math
|
geometry_perpendicular_intersection_train_0029
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 3, from vertex 2 to vertex 10, from vertex 2 to vertex 9, from vertex 10 to vertex 7, from vertex 5 to vertex 2, from vertex 1 to vertex 8, from vertex 7 to vertex 3, from vertex 9 to vertex 3, from vertex 2 to vertex 6, from vertex 10 to vertex 4, from vertex 10 to vertex 5, from vertex 7 to vertex 1, and from vertex 4 to vertex 8. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
9
|
math
|
geometry_perpendicular_intersection_train_0030
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 11-gon, with vertices numbered 1 through 11 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 7, from vertex 7 to vertex 11, from vertex 2 to vertex 10, from vertex 2 to vertex 4, from vertex 4 to vertex 10, from vertex 10 to vertex 6, from vertex 7 to vertex 9, from vertex 9 to vertex 5, from vertex 6 to vertex 1, from vertex 7 to vertex 4, from vertex 9 to vertex 3, and from vertex 3 to vertex 10. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
0
|
math
|
geometry_perpendicular_intersection_train_0031
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 7-gon, with vertices numbered 1 through 7 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 1, from vertex 6 to vertex 3, from vertex 7 to vertex 3, from vertex 5 to vertex 3, from vertex 5 to vertex 2, from vertex 2 to vertex 6, from vertex 1 to vertex 4, and from vertex 2 to vertex 7. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
0
|
math
|
geometry_perpendicular_intersection_train_0032
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 6, from vertex 2 to vertex 5, from vertex 6 to vertex 2, from vertex 2 to vertex 4, from vertex 3 to vertex 1, from vertex 5 to vertex 3, from vertex 4 to vertex 1, and from vertex 5 to vertex 1. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
4
|
math
|
geometry_perpendicular_intersection_train_0033
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 7 to vertex 2, from vertex 1 to vertex 3, from vertex 5 to vertex 8, from vertex 2 to vertex 4, from vertex 7 to vertex 4, from vertex 3 to vertex 8, from vertex 4 to vertex 1, from vertex 3 to vertex 7, from vertex 3 to vertex 5, from vertex 8 to vertex 4, and from vertex 2 to vertex 6. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
7
|
math
|
geometry_perpendicular_intersection_train_0034
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 4, from vertex 7 to vertex 10, from vertex 10 to vertex 4, from vertex 8 to vertex 5, from vertex 6 to vertex 10, from vertex 3 to vertex 1, from vertex 8 to vertex 2, from vertex 1 to vertex 8, from vertex 10 to vertex 5, from vertex 9 to vertex 6, from vertex 3 to vertex 6, and from vertex 5 to vertex 2. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
6
|
math
|
geometry_perpendicular_intersection_train_0035
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 3, from vertex 3 to vertex 8, from vertex 4 to vertex 6, from vertex 11 to vertex 7, from vertex 5 to vertex 10, from vertex 9 to vertex 1, from vertex 3 to vertex 7, from vertex 2 to vertex 6, from vertex 10 to vertex 6, from vertex 12 to vertex 4, and from vertex 4 to vertex 9. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
9
|
math
|
geometry_perpendicular_intersection_train_0036
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 12 to vertex 10, from vertex 6 to vertex 2, from vertex 9 to vertex 11, from vertex 2 to vertex 10, from vertex 1 to vertex 10, from vertex 5 to vertex 7, from vertex 5 to vertex 11, from vertex 2 to vertex 11, from vertex 1 to vertex 5, from vertex 7 to vertex 9, from vertex 5 to vertex 12, and from vertex 6 to vertex 10. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
6
|
math
|
geometry_perpendicular_intersection_train_0037
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 3, from vertex 1 to vertex 8, from vertex 8 to vertex 2, from vertex 5 to vertex 10, from vertex 9 to vertex 12, from vertex 1 to vertex 4, from vertex 11 to vertex 5, from vertex 11 to vertex 7, and from vertex 2 to vertex 4. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
4
|
math
|
geometry_perpendicular_intersection_train_0038
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 3, from vertex 7 to vertex 5, from vertex 2 to vertex 6, from vertex 4 to vertex 7, from vertex 6 to vertex 4, from vertex 4 to vertex 8, from vertex 1 to vertex 7, from vertex 7 to vertex 2, from vertex 8 to vertex 3, from vertex 1 to vertex 4, and from vertex 5 to vertex 8. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
8
|
math
|
geometry_perpendicular_intersection_train_0039
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 6, from vertex 7 to vertex 3, from vertex 1 to vertex 4, from vertex 3 to vertex 1, from vertex 5 to vertex 1, from vertex 7 to vertex 5, from vertex 6 to vertex 8, from vertex 8 to vertex 5, from vertex 4 to vertex 2, from vertex 3 to vertex 6, and from vertex 2 to vertex 6. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
10
|
math
|
geometry_perpendicular_intersection_train_0040
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 10, from vertex 9 to vertex 5, from vertex 12 to vertex 6, from vertex 8 to vertex 10, from vertex 6 to vertex 3, from vertex 12 to vertex 2, from vertex 2 to vertex 9, from vertex 10 to vertex 12, from vertex 9 to vertex 3, from vertex 7 to vertex 2, and from vertex 11 to vertex 3. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
4
|
math
|
geometry_perpendicular_intersection_train_0041
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 1, from vertex 1 to vertex 7, from vertex 10 to vertex 7, from vertex 2 to vertex 5, from vertex 5 to vertex 3, from vertex 6 to vertex 3, from vertex 4 to vertex 9, from vertex 8 to vertex 4, from vertex 7 to vertex 3, from vertex 6 to vertex 9, from vertex 1 to vertex 8, from vertex 9 to vertex 7, and from vertex 3 to vertex 10. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
9
|
math
|
geometry_perpendicular_intersection_train_0042
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 5, from vertex 4 to vertex 1, from vertex 6 to vertex 4, from vertex 3 to vertex 1, from vertex 1 to vertex 5, from vertex 6 to vertex 3, from vertex 6 to vertex 2, and from vertex 3 to vertex 5. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
5
|
math
|
geometry_perpendicular_intersection_train_0043
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 7-gon, with vertices numbered 1 through 7 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 6, from vertex 2 to vertex 6, from vertex 4 to vertex 7, from vertex 3 to vertex 7, from vertex 7 to vertex 5, from vertex 5 to vertex 1, from vertex 2 to vertex 5, and from vertex 1 to vertex 6. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
0
|
math
|
geometry_perpendicular_intersection_train_0044
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 5, from vertex 8 to vertex 4, from vertex 1 to vertex 7, from vertex 7 to vertex 5, from vertex 3 to vertex 6, from vertex 1 to vertex 3, from vertex 6 to vertex 2, from vertex 7 to vertex 2, and from vertex 5 to vertex 3. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
9
|
math
|
geometry_perpendicular_intersection_train_0045
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 6, from vertex 8 to vertex 3, from vertex 8 to vertex 5, from vertex 7 to vertex 3, from vertex 3 to vertex 1, from vertex 8 to vertex 4, from vertex 1 to vertex 6, from vertex 2 to vertex 4, from vertex 8 to vertex 2, from vertex 5 to vertex 7, from vertex 2 to vertex 7, and from vertex 4 to vertex 6. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
8
|
math
|
geometry_perpendicular_intersection_train_0046
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 7, from vertex 6 to vertex 8, from vertex 6 to vertex 4, from vertex 4 to vertex 1, from vertex 4 to vertex 8, from vertex 6 to vertex 3, from vertex 8 to vertex 3, from vertex 2 to vertex 7, from vertex 6 to vertex 1, from vertex 3 to vertex 1, and from vertex 3 to vertex 5. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
7
|
math
|
geometry_perpendicular_intersection_train_0047
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 7 to vertex 9, from vertex 4 to vertex 12, from vertex 7 to vertex 5, from vertex 6 to vertex 8, from vertex 9 to vertex 1, from vertex 2 to vertex 10, and from vertex 5 to vertex 1. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
4
|
math
|
geometry_perpendicular_intersection_train_0048
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 3, from vertex 7 to vertex 3, from vertex 4 to vertex 8, from vertex 7 to vertex 4, from vertex 5 to vertex 3, from vertex 4 to vertex 2, from vertex 5 to vertex 8, from vertex 6 to vertex 2, from vertex 5 to vertex 2, from vertex 8 to vertex 2, and from vertex 1 to vertex 6. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
8
|
math
|
geometry_perpendicular_intersection_train_0049
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 8, from vertex 1 to vertex 7, from vertex 4 to vertex 6, from vertex 3 to vertex 5, from vertex 8 to vertex 4, from vertex 3 to vertex 1, and from vertex 1 to vertex 4. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
4
|
math
|
geometry_perpendicular_intersection_train_0050
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 11 to vertex 5, from vertex 10 to vertex 4, from vertex 5 to vertex 2, from vertex 1 to vertex 10, from vertex 2 to vertex 11, from vertex 8 to vertex 3, from vertex 8 to vertex 1, from vertex 9 to vertex 1, from vertex 4 to vertex 6, from vertex 3 to vertex 12, from vertex 12 to vertex 2, from vertex 5 to vertex 8, and from vertex 5 to vertex 12. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
7
|
math
|
geometry_perpendicular_intersection_train_0051
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 6, from vertex 10 to vertex 3, from vertex 6 to vertex 8, from vertex 6 to vertex 10, from vertex 9 to vertex 2, from vertex 6 to vertex 9, from vertex 10 to vertex 2, from vertex 5 to vertex 1, from vertex 12 to vertex 3, from vertex 1 to vertex 10, from vertex 8 to vertex 11, and from vertex 10 to vertex 4. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
3
|
math
|
geometry_perpendicular_intersection_train_0052
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 9 to vertex 2, from vertex 10 to vertex 1, from vertex 12 to vertex 8, from vertex 6 to vertex 9, from vertex 6 to vertex 12, from vertex 12 to vertex 5, and from vertex 1 to vertex 4. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
4
|
math
|
geometry_perpendicular_intersection_train_0053
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 3, from vertex 4 to vertex 2, from vertex 4 to vertex 7, from vertex 5 to vertex 1, from vertex 8 to vertex 3, from vertex 1 to vertex 7, from vertex 3 to vertex 6, from vertex 6 to vertex 2, from vertex 8 to vertex 4, and from vertex 2 to vertex 7. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
4
|
math
|
geometry_perpendicular_intersection_train_0054
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 5, from vertex 1 to vertex 4, from vertex 3 to vertex 5, from vertex 2 to vertex 4, from vertex 5 to vertex 1, from vertex 1 to vertex 3, from vertex 6 to vertex 2, from vertex 6 to vertex 4, and from vertex 6 to vertex 3. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
6
|
math
|
geometry_perpendicular_intersection_train_0055
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 9, from vertex 10 to vertex 6, from vertex 11 to vertex 6, from vertex 3 to vertex 1, from vertex 5 to vertex 1, from vertex 6 to vertex 3, from vertex 10 to vertex 1, from vertex 7 to vertex 1, from vertex 10 to vertex 2, from vertex 4 to vertex 6, from vertex 3 to vertex 7, from vertex 1 to vertex 6, and from vertex 5 to vertex 8. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
9
|
math
|
geometry_perpendicular_intersection_train_0056
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 7 to vertex 9, from vertex 7 to vertex 1, from vertex 3 to vertex 1, from vertex 1 to vertex 9, from vertex 5 to vertex 1, from vertex 9 to vertex 6, from vertex 3 to vertex 9, from vertex 1 to vertex 6, from vertex 10 to vertex 5, from vertex 5 to vertex 3, from vertex 9 to vertex 2, from vertex 8 to vertex 5, and from vertex 10 to vertex 6. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
8
|
math
|
geometry_perpendicular_intersection_train_0057
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 1, from vertex 3 to vertex 6, from vertex 5 to vertex 3, from vertex 6 to vertex 2, from vertex 6 to vertex 4, from vertex 5 to vertex 1, from vertex 2 to vertex 4, and from vertex 2 to vertex 5. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
5
|
math
|
geometry_perpendicular_intersection_train_0058
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 9, from vertex 5 to vertex 1, from vertex 7 to vertex 9, from vertex 7 to vertex 4, from vertex 9 to vertex 3, from vertex 6 to vertex 9, from vertex 3 to vertex 1, from vertex 3 to vertex 10, from vertex 1 to vertex 9, and from vertex 4 to vertex 6. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
4
|
math
|
geometry_perpendicular_intersection_train_0059
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 11, from vertex 5 to vertex 1, from vertex 6 to vertex 10, from vertex 6 to vertex 12, from vertex 8 to vertex 12, from vertex 2 to vertex 12, from vertex 12 to vertex 3, from vertex 9 to vertex 2, from vertex 2 to vertex 11, and from vertex 12 to vertex 10. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
3
|
math
|
geometry_perpendicular_intersection_train_0060
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 11 to vertex 1, from vertex 9 to vertex 5, from vertex 2 to vertex 11, from vertex 4 to vertex 1, from vertex 12 to vertex 8, from vertex 2 to vertex 8, from vertex 2 to vertex 6, from vertex 6 to vertex 12, from vertex 10 to vertex 8, from vertex 9 to vertex 12, from vertex 7 to vertex 9, from vertex 6 to vertex 9, and from vertex 6 to vertex 3. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
7
|
math
|
geometry_perpendicular_intersection_train_0061
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 11 to vertex 2, from vertex 8 to vertex 10, from vertex 8 to vertex 1, from vertex 7 to vertex 4, from vertex 3 to vertex 11, from vertex 1 to vertex 4, from vertex 10 to vertex 7, and from vertex 5 to vertex 9. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
2
|
math
|
geometry_perpendicular_intersection_train_0062
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 2, from vertex 8 to vertex 4, from vertex 1 to vertex 4, from vertex 1 to vertex 7, from vertex 7 to vertex 5, from vertex 5 to vertex 1, from vertex 7 to vertex 4, from vertex 5 to vertex 3, from vertex 5 to vertex 2, from vertex 6 to vertex 8, and from vertex 6 to vertex 1. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
6
|
math
|
geometry_perpendicular_intersection_train_0063
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 3, from vertex 6 to vertex 4, from vertex 5 to vertex 3, from vertex 5 to vertex 1, from vertex 2 to vertex 6, from vertex 4 to vertex 1, from vertex 6 to vertex 3, from vertex 2 to vertex 4, and from vertex 2 to vertex 5. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
6
|
math
|
geometry_perpendicular_intersection_train_0064
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 8, from vertex 4 to vertex 8, from vertex 4 to vertex 1, from vertex 3 to vertex 8, from vertex 1 to vertex 5, from vertex 6 to vertex 3, from vertex 7 to vertex 4, from vertex 1 to vertex 6, and from vertex 8 to vertex 2. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
5
|
math
|
geometry_perpendicular_intersection_train_0065
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 9, from vertex 4 to vertex 1, from vertex 5 to vertex 3, from vertex 2 to vertex 10, from vertex 5 to vertex 2, from vertex 7 to vertex 2, and from vertex 1 to vertex 5. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
1
|
math
|
geometry_perpendicular_intersection_train_0066
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 8 to vertex 6, from vertex 7 to vertex 4, from vertex 3 to vertex 1, from vertex 8 to vertex 3, from vertex 3 to vertex 6, from vertex 4 to vertex 8, from vertex 5 to vertex 8, from vertex 6 to vertex 4, from vertex 2 to vertex 5, and from vertex 2 to vertex 7. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
5
|
math
|
geometry_perpendicular_intersection_train_0067
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 7 to vertex 4, from vertex 4 to vertex 6, from vertex 3 to vertex 8, from vertex 2 to vertex 5, from vertex 1 to vertex 7, from vertex 3 to vertex 6, from vertex 3 to vertex 5, from vertex 5 to vertex 1, from vertex 6 to vertex 2, from vertex 4 to vertex 1, and from vertex 7 to vertex 5. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
7
|
math
|
geometry_perpendicular_intersection_train_0068
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 8, from vertex 1 to vertex 5, from vertex 2 to vertex 6, from vertex 8 to vertex 4, from vertex 3 to vertex 5, from vertex 8 to vertex 6, from vertex 4 to vertex 2, and from vertex 4 to vertex 1. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
5
|
math
|
geometry_perpendicular_intersection_train_0069
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 8, from vertex 7 to vertex 1, from vertex 12 to vertex 3, from vertex 12 to vertex 7, from vertex 4 to vertex 6, from vertex 12 to vertex 2, from vertex 3 to vertex 6, from vertex 11 to vertex 5, from vertex 10 to vertex 3, from vertex 10 to vertex 7, from vertex 5 to vertex 12, from vertex 12 to vertex 8, and from vertex 12 to vertex 9. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
7
|
math
|
geometry_perpendicular_intersection_train_0070
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 4, from vertex 5 to vertex 7, from vertex 1 to vertex 7, from vertex 4 to vertex 2, from vertex 6 to vertex 2, from vertex 8 to vertex 6, and from vertex 1 to vertex 4. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
4
|
math
|
geometry_perpendicular_intersection_train_0071
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 9, from vertex 4 to vertex 12, from vertex 8 to vertex 5, from vertex 2 to vertex 7, from vertex 5 to vertex 11, from vertex 4 to vertex 7, from vertex 7 to vertex 12, from vertex 5 to vertex 3, from vertex 8 to vertex 4, from vertex 2 to vertex 11, from vertex 1 to vertex 7, from vertex 2 to vertex 6, and from vertex 5 to vertex 7. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
4
|
math
|
geometry_perpendicular_intersection_train_0072
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 3, from vertex 7 to vertex 2, from vertex 6 to vertex 2, from vertex 8 to vertex 4, from vertex 12 to vertex 10, from vertex 5 to vertex 7, from vertex 9 to vertex 1, from vertex 10 to vertex 7, from vertex 8 to vertex 10, from vertex 11 to vertex 7, from vertex 7 to vertex 4, from vertex 5 to vertex 9, and from vertex 3 to vertex 7. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
9
|
math
|
geometry_perpendicular_intersection_train_0073
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 9, from vertex 12 to vertex 2, from vertex 11 to vertex 8, from vertex 6 to vertex 12, from vertex 8 to vertex 6, from vertex 7 to vertex 3, from vertex 11 to vertex 4, from vertex 7 to vertex 2, from vertex 3 to vertex 1, from vertex 3 to vertex 6, from vertex 4 to vertex 9, and from vertex 10 to vertex 2. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
6
|
math
|
geometry_perpendicular_intersection_train_0074
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 10 to vertex 8, from vertex 2 to vertex 4, from vertex 3 to vertex 8, from vertex 3 to vertex 9, from vertex 4 to vertex 6, from vertex 1 to vertex 11, from vertex 12 to vertex 10, from vertex 2 to vertex 7, from vertex 11 to vertex 6, from vertex 11 to vertex 9, and from vertex 9 to vertex 4. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
5
|
math
|
geometry_perpendicular_intersection_train_0075
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 10, from vertex 11 to vertex 6, from vertex 5 to vertex 10, from vertex 7 to vertex 3, from vertex 7 to vertex 2, from vertex 5 to vertex 8, from vertex 4 to vertex 12, from vertex 8 to vertex 1, from vertex 6 to vertex 3, from vertex 6 to vertex 1, from vertex 7 to vertex 1, and from vertex 2 to vertex 4. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
6
|
math
|
geometry_perpendicular_intersection_train_0076
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 7 to vertex 1, from vertex 8 to vertex 5, from vertex 2 to vertex 4, from vertex 9 to vertex 2, from vertex 9 to vertex 5, from vertex 7 to vertex 4, from vertex 7 to vertex 10, from vertex 6 to vertex 9, from vertex 8 to vertex 4, from vertex 10 to vertex 5, from vertex 7 to vertex 5, and from vertex 6 to vertex 1. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
7
|
math
|
geometry_perpendicular_intersection_train_0077
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 8 to vertex 3, from vertex 2 to vertex 12, from vertex 2 to vertex 6, from vertex 10 to vertex 3, from vertex 2 to vertex 10, from vertex 2 to vertex 5, from vertex 1 to vertex 4, from vertex 10 to vertex 7, and from vertex 5 to vertex 8. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
5
|
math
|
geometry_perpendicular_intersection_train_0078
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 2, from vertex 11 to vertex 2, from vertex 5 to vertex 9, from vertex 10 to vertex 3, from vertex 6 to vertex 8, from vertex 8 to vertex 1, from vertex 2 to vertex 7, from vertex 9 to vertex 7, from vertex 11 to vertex 8, from vertex 8 to vertex 3, from vertex 6 to vertex 12, and from vertex 5 to vertex 3. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
6
|
math
|
geometry_perpendicular_intersection_train_0079
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 11-gon, with vertices numbered 1 through 11 in counterclockwise order. Draw the diagonal from vertex 10 to vertex 7, from vertex 11 to vertex 4, from vertex 7 to vertex 4, from vertex 5 to vertex 2, from vertex 10 to vertex 5, from vertex 8 to vertex 4, from vertex 3 to vertex 8, and from vertex 9 to vertex 1. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
0
|
math
|
geometry_perpendicular_intersection_train_0080
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 1, from vertex 2 to vertex 7, from vertex 8 to vertex 2, from vertex 4 to vertex 1, from vertex 7 to vertex 5, from vertex 8 to vertex 4, from vertex 5 to vertex 2, from vertex 8 to vertex 6, from vertex 6 to vertex 2, and from vertex 3 to vertex 5. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
8
|
math
|
geometry_perpendicular_intersection_train_0081
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 2, from vertex 4 to vertex 9, from vertex 4 to vertex 11, from vertex 7 to vertex 5, from vertex 8 to vertex 12, from vertex 6 to vertex 8, from vertex 3 to vertex 11, from vertex 4 to vertex 10, and from vertex 3 to vertex 6. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
5
|
math
|
geometry_perpendicular_intersection_train_0082
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 2, from vertex 7 to vertex 9, from vertex 1 to vertex 5, from vertex 6 to vertex 9, from vertex 7 to vertex 1, from vertex 1 to vertex 8, and from vertex 2 to vertex 10. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
1
|
math
|
geometry_perpendicular_intersection_train_0083
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 10 to vertex 5, from vertex 2 to vertex 6, from vertex 1 to vertex 7, from vertex 6 to vertex 10, from vertex 6 to vertex 4, from vertex 8 to vertex 2, from vertex 8 to vertex 1, from vertex 5 to vertex 8, from vertex 9 to vertex 4, and from vertex 2 to vertex 9. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
7
|
math
|
geometry_perpendicular_intersection_train_0084
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 7 to vertex 4, from vertex 6 to vertex 9, from vertex 8 to vertex 6, from vertex 9 to vertex 1, from vertex 3 to vertex 9, from vertex 1 to vertex 7, from vertex 4 to vertex 1, from vertex 6 to vertex 10, from vertex 5 to vertex 8, from vertex 3 to vertex 7, from vertex 2 to vertex 6, and from vertex 9 to vertex 7. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
8
|
math
|
geometry_perpendicular_intersection_train_0085
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 6, from vertex 3 to vertex 6, from vertex 6 to vertex 4, from vertex 3 to vertex 1, from vertex 3 to vertex 5, from vertex 1 to vertex 4, from vertex 2 to vertex 4, and from vertex 1 to vertex 5. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
4
|
math
|
geometry_perpendicular_intersection_train_0086
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 10, from vertex 5 to vertex 9, from vertex 1 to vertex 8, from vertex 9 to vertex 2, from vertex 6 to vertex 4, from vertex 1 to vertex 4, from vertex 2 to vertex 4, from vertex 6 to vertex 2, from vertex 10 to vertex 7, from vertex 10 to vertex 2, from vertex 10 to vertex 3, from vertex 5 to vertex 1, and from vertex 6 to vertex 3. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
9
|
math
|
geometry_perpendicular_intersection_train_0087
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 4, from vertex 7 to vertex 4, from vertex 11 to vertex 4, from vertex 6 to vertex 10, from vertex 2 to vertex 12, from vertex 6 to vertex 8, from vertex 3 to vertex 11, from vertex 11 to vertex 9, from vertex 11 to vertex 8, from vertex 1 to vertex 11, and from vertex 7 to vertex 3. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
5
|
math
|
geometry_perpendicular_intersection_train_0088
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 5, from vertex 2 to vertex 5, from vertex 8 to vertex 6, from vertex 3 to vertex 7, from vertex 6 to vertex 4, from vertex 3 to vertex 6, from vertex 7 to vertex 4, and from vertex 8 to vertex 3. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
4
|
math
|
geometry_perpendicular_intersection_train_0089
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 6, from vertex 2 to vertex 4, from vertex 1 to vertex 5, from vertex 5 to vertex 2, from vertex 1 to vertex 3, from vertex 6 to vertex 3, and from vertex 5 to vertex 3. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
3
|
math
|
geometry_perpendicular_intersection_train_0090
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 7, from vertex 1 to vertex 6, from vertex 3 to vertex 9, from vertex 5 to vertex 10, from vertex 3 to vertex 6, from vertex 5 to vertex 7, from vertex 2 to vertex 5, from vertex 2 to vertex 7, from vertex 9 to vertex 6, from vertex 5 to vertex 3, and from vertex 2 to vertex 4. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
4
|
math
|
geometry_perpendicular_intersection_train_0091
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 7, from vertex 4 to vertex 7, from vertex 6 to vertex 1, from vertex 2 to vertex 5, from vertex 1 to vertex 7, from vertex 8 to vertex 5, from vertex 5 to vertex 1, from vertex 4 to vertex 8, from vertex 2 to vertex 6, from vertex 7 to vertex 2, and from vertex 6 to vertex 8. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
7
|
math
|
geometry_perpendicular_intersection_train_0092
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 8, from vertex 7 to vertex 5, from vertex 2 to vertex 5, from vertex 4 to vertex 2, from vertex 3 to vertex 6, from vertex 5 to vertex 3, from vertex 5 to vertex 1, from vertex 8 to vertex 4, from vertex 1 to vertex 3, from vertex 3 to vertex 7, from vertex 2 to vertex 6, from vertex 4 to vertex 7, and from vertex 7 to vertex 1. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
13
|
math
|
geometry_perpendicular_intersection_train_0093
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 9, from vertex 3 to vertex 8, from vertex 4 to vertex 1, from vertex 5 to vertex 7, from vertex 1 to vertex 7, from vertex 5 to vertex 8, from vertex 10 to vertex 3, and from vertex 2 to vertex 7. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
3
|
math
|
geometry_perpendicular_intersection_train_0094
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 3, from vertex 6 to vertex 3, from vertex 4 to vertex 1, from vertex 2 to vertex 6, from vertex 4 to vertex 2, from vertex 4 to vertex 6, and from vertex 1 to vertex 5. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
4
|
math
|
geometry_perpendicular_intersection_train_0095
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 5, from vertex 4 to vertex 2, from vertex 1 to vertex 3, from vertex 2 to vertex 5, from vertex 6 to vertex 2, from vertex 1 to vertex 4, and from vertex 6 to vertex 4. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
4
|
math
|
geometry_perpendicular_intersection_train_0096
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 8 to vertex 4, from vertex 4 to vertex 2, from vertex 5 to vertex 9, from vertex 6 to vertex 12, from vertex 9 to vertex 7, from vertex 8 to vertex 6, from vertex 1 to vertex 5, from vertex 4 to vertex 9, and from vertex 11 to vertex 1. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
6
|
math
|
geometry_perpendicular_intersection_train_0097
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 11, from vertex 3 to vertex 7, from vertex 7 to vertex 2, from vertex 3 to vertex 6, from vertex 10 to vertex 3, from vertex 4 to vertex 1, and from vertex 8 to vertex 4. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
0
|
math
|
geometry_perpendicular_intersection_train_0098
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 10 to vertex 1, from vertex 11 to vertex 6, from vertex 5 to vertex 2, from vertex 1 to vertex 5, from vertex 10 to vertex 5, from vertex 1 to vertex 9, from vertex 8 to vertex 2, from vertex 7 to vertex 5, from vertex 11 to vertex 9, from vertex 12 to vertex 10, and from vertex 8 to vertex 12. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
2
|
math
|
geometry_perpendicular_intersection_train_0099
|
geometry_perpendicular_intersection
| null |
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 11 to vertex 4, from vertex 9 to vertex 11, from vertex 11 to vertex 5, from vertex 10 to vertex 2, from vertex 5 to vertex 12, from vertex 11 to vertex 7, from vertex 3 to vertex 8, and from vertex 11 to vertex 3. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
3
|
math
|
End of preview. Expand
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