# Geometry Foundations

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20260912-geometry-foundations · 2026-09-12 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Coordinate-Plane Distance

In the Euclidean plane, P=(-23, -26) and Q=(-20, -22). Find the distance d(P,Q).

View answer and explanation5

The coordinate difference is (3, 4), so d=√((3)²+(4)²)=5.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-distance-coordinate-0001/)

Euclidean length · qf-distance-coordinate · candidate





## 2. Coordinate-Plane Distance

In the Euclidean plane, P=(-16, -15) and Q=(-21, -3). Find the distance d(P,Q).

View answer and explanation13

The coordinate difference is (-5, 12), so d=√((-5)²+(12)²)=13.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-distance-coordinate-0002/)

Euclidean length · qf-distance-coordinate · candidate





## 3. Coordinate-Plane Distance

In the Euclidean plane, P=(-9, -4) and Q=(-1, -19). Find the distance d(P,Q).

View answer and explanation17

The coordinate difference is (8, -15), so d=√((8)²+(-15)²)=17.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-distance-coordinate-0003/)

Euclidean length · qf-distance-coordinate · candidate





## 4. Coordinate-Plane Distance

In the Euclidean plane, P=(-2, 7) and Q=(-9, -17). Find the distance d(P,Q).

View answer and explanation25

The coordinate difference is (-7, -24), so d=√((-7)²+(-24)²)=25.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-distance-coordinate-0004/)

Euclidean length · qf-distance-coordinate · candidate





## 5. Coordinate-Plane Distance

In the Euclidean plane, P=(5, 18) and Q=(25, 39). Find the distance d(P,Q).

View answer and explanation29

The coordinate difference is (20, 21), so d=√((20)²+(21)²)=29.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-distance-coordinate-0005/)

Euclidean length · qf-distance-coordinate · candidate





## 6. Coordinate-Plane Distance

In the Euclidean plane, P=(12, -24) and Q=(3, 16). Find the distance d(P,Q).

View answer and explanation41

The coordinate difference is (-9, 40), so d=√((-9)²+(40)²)=41.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-distance-coordinate-0006/)

Euclidean length · qf-distance-coordinate · candidate





## 7. Coordinate-Plane Distance

In the Euclidean plane, P=(19, -13) and Q=(31, -48). Find the distance d(P,Q).

View answer and explanation37

The coordinate difference is (12, -35), so d=√((12)²+(-35)²)=37.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-distance-coordinate-0007/)

Euclidean length · qf-distance-coordinate · candidate





## 8. Coordinate-Plane Distance

In the Euclidean plane, P=(-21, -2) and Q=(-32, -62). Find the distance d(P,Q).

View answer and explanation61

The coordinate difference is (-11, -60), so d=√((-11)²+(-60)²)=61.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-distance-coordinate-0008/)

Euclidean length · qf-distance-coordinate · candidate





## 9. Coordinate-Plane Distance

In the Euclidean plane, P=(-14, 9) and Q=(14, 54). Find the distance d(P,Q).

View answer and explanation53

The coordinate difference is (28, 45), so d=√((28)²+(45)²)=53.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-distance-coordinate-0009/)

Euclidean length · qf-distance-coordinate · candidate





## 10. Coordinate-Plane Distance

In the Euclidean plane, P=(-7, 20) and Q=(-40, 76). Find the distance d(P,Q).

View answer and explanation65

The coordinate difference is (-33, 56), so d=√((-33)²+(56)²)=65.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-distance-coordinate-0010/)

Euclidean length · qf-distance-coordinate · candidate





## 11. Coordinate-Plane Distance

In the Euclidean plane, P=(0, -22) and Q=(16, -85). Find the distance d(P,Q).

View answer and explanation65

The coordinate difference is (16, -63), so d=√((16)²+(-63)²)=65.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-distance-coordinate-0011/)

Euclidean length · qf-distance-coordinate · candidate





## 12. Coordinate-Plane Distance

In the Euclidean plane, P=(7, -11) and Q=(-41, -66). Find the distance d(P,Q).

View answer and explanation73

The coordinate difference is (-48, -55), so d=√((-48)²+(-55)²)=73.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-distance-coordinate-0012/)

Euclidean length · qf-distance-coordinate · candidate





## 13. Coordinate-Plane Distance

In the Euclidean plane, P=(14, 0) and Q=(27, 84). Find the distance d(P,Q).

View answer and explanation85

The coordinate difference is (13, 84), so d=√((13)²+(84)²)=85.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-distance-coordinate-0013/)

Euclidean length · qf-distance-coordinate · candidate





## 14. Coordinate-Plane Distance

In the Euclidean plane, P=(21, 11) and Q=(-15, 88). Find the distance d(P,Q).

View answer and explanation85

The coordinate difference is (-36, 77), so d=√((-36)²+(77)²)=85.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-distance-coordinate-0014/)

Euclidean length · qf-distance-coordinate · candidate





## 15. Coordinate-Plane Distance

In the Euclidean plane, P=(-19, 22) and Q=(20, -58). Find the distance d(P,Q).

View answer and explanation89

The coordinate difference is (39, -80), so d=√((39)²+(-80)²)=89.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-distance-coordinate-0015/)

Euclidean length · qf-distance-coordinate · candidate





## 16. Coordinate-Plane Distance

In the Euclidean plane, P=(-12, -20) and Q=(-77, -92). Find the distance d(P,Q).

View answer and explanation97

The coordinate difference is (-65, -72), so d=√((-65)²+(-72)²)=97.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-distance-coordinate-0016/)

Euclidean length · qf-distance-coordinate · candidate





## 17. Coordinate-Plane Distance

In the Euclidean plane, P=(-5, -9) and Q=(15, 90). Find the distance d(P,Q).

View answer and explanation101

The coordinate difference is (20, 99), so d=√((20)²+(99)²)=101.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-distance-coordinate-0017/)

Euclidean length · qf-distance-coordinate · candidate





## 18. Coordinate-Plane Distance

In the Euclidean plane, P=(2, 2) and Q=(-58, 93). Find the distance d(P,Q).

View answer and explanation109

The coordinate difference is (-60, 91), so d=√((-60)²+(91)²)=109.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-distance-coordinate-0018/)

Euclidean length · qf-distance-coordinate · candidate





## 19. Coordinate-Plane Distance

In the Euclidean plane, P=(9, 13) and Q=(24, -99). Find the distance d(P,Q).

View answer and explanation113

The coordinate difference is (15, -112), so d=√((15)²+(-112)²)=113.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-distance-coordinate-0019/)

Euclidean length · qf-distance-coordinate · candidate





## 20. Coordinate-Plane Distance

In the Euclidean plane, P=(16, 24) and Q=(-28, -93). Find the distance d(P,Q).

View answer and explanation125

The coordinate difference is (-44, -117), so d=√((-44)²+(-117)²)=125.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-distance-coordinate-0020/)

Euclidean length · qf-distance-coordinate · candidate





## 21. Coordinate-Plane Distance

In the Euclidean plane, P=(23, -18) and Q=(29, -10). Find the distance d(P,Q).

View answer and explanation10

The coordinate difference is (6, 8), so d=√((6)²+(8)²)=10.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-distance-coordinate-0021/)

Euclidean length · qf-distance-coordinate · candidate





## 22. Coordinate-Plane Distance

In the Euclidean plane, P=(-17, -7) and Q=(-27, 17). Find the distance d(P,Q).

View answer and explanation26

The coordinate difference is (-10, 24), so d=√((-10)²+(24)²)=26.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-distance-coordinate-0022/)

Euclidean length · qf-distance-coordinate · candidate





## 23. Coordinate-Plane Distance

In the Euclidean plane, P=(-10, 4) and Q=(6, -26). Find the distance d(P,Q).

View answer and explanation34

The coordinate difference is (16, -30), so d=√((16)²+(-30)²)=34.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-distance-coordinate-0023/)

Euclidean length · qf-distance-coordinate · candidate





## 24. Coordinate-Plane Distance

In the Euclidean plane, P=(-3, 15) and Q=(-17, -33). Find the distance d(P,Q).

View answer and explanation50

The coordinate difference is (-14, -48), so d=√((-14)²+(-48)²)=50.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-distance-coordinate-0024/)

Euclidean length · qf-distance-coordinate · candidate





## 25. Coordinate-Plane Distance

In the Euclidean plane, P=(4, 26) and Q=(44, 68). Find the distance d(P,Q).

View answer and explanation58

The coordinate difference is (40, 42), so d=√((40)²+(42)²)=58.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-distance-coordinate-0025/)

Euclidean length · qf-distance-coordinate · candidate





## 26. Coordinate-Plane Distance

In the Euclidean plane, P=(11, -16) and Q=(-7, 64). Find the distance d(P,Q).

View answer and explanation82

The coordinate difference is (-18, 80), so d=√((-18)²+(80)²)=82.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-distance-coordinate-0026/)

Euclidean length · qf-distance-coordinate · candidate





## 27. Coordinate-Plane Distance

In the Euclidean plane, P=(18, -5) and Q=(42, -75). Find the distance d(P,Q).

View answer and explanation74

The coordinate difference is (24, -70), so d=√((24)²+(-70)²)=74.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-distance-coordinate-0027/)

Euclidean length · qf-distance-coordinate · candidate





## 28. Coordinate-Plane Distance

In the Euclidean plane, P=(-22, 6) and Q=(-44, -114). Find the distance d(P,Q).

View answer and explanation122

The coordinate difference is (-22, -120), so d=√((-22)²+(-120)²)=122.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-distance-coordinate-0028/)

Euclidean length · qf-distance-coordinate · candidate





## 29. Coordinate-Plane Distance

In the Euclidean plane, P=(-15, 17) and Q=(41, 107). Find the distance d(P,Q).

View answer and explanation106

The coordinate difference is (56, 90), so d=√((56)²+(90)²)=106.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-distance-coordinate-0029/)

Euclidean length · qf-distance-coordinate · candidate





## 30. Coordinate-Plane Distance

In the Euclidean plane, P=(-8, -25) and Q=(-74, 87). Find the distance d(P,Q).

View answer and explanation130

The coordinate difference is (-66, 112), so d=√((-66)²+(112)²)=130.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-distance-coordinate-0030/)

Euclidean length · qf-distance-coordinate · candidate





## 31. Identifying a Right Triangle

A nondegenerate triangle has interior angles 10°, 90°, 80°. Is it a right triangle?

View answer and explanationYes

The angles sum to 180° and one angle is 90°, so the triangle is right.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-right-triangle-determination-0001/)

Right triangle · qf-right-triangle-determination · candidate





## 32. Identifying a Right Triangle

A nondegenerate triangle has interior angles 15°, 80°, 85°. Is it a right triangle?

View answer and explanationNo

Although the angles sum to 180°, none is 90°, so the triangle is not right.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-right-triangle-determination-0002/)

Right triangle · qf-right-triangle-determination · candidate





## 33. Identifying a Right Triangle

A nondegenerate triangle has interior angles 20°, 90°, 70°. Is it a right triangle?

View answer and explanationYes

The angles sum to 180° and one angle is 90°, so the triangle is right.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-right-triangle-determination-0003/)

Right triangle · qf-right-triangle-determination · candidate





## 34. Identifying a Right Triangle

A nondegenerate triangle has interior angles 25°, 80°, 75°. Is it a right triangle?

View answer and explanationNo

Although the angles sum to 180°, none is 90°, so the triangle is not right.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-right-triangle-determination-0004/)

Right triangle · qf-right-triangle-determination · candidate





## 35. Identifying a Right Triangle

A nondegenerate triangle has interior angles 30°, 90°, 60°. Is it a right triangle?

View answer and explanationYes

The angles sum to 180° and one angle is 90°, so the triangle is right.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-right-triangle-determination-0005/)

Right triangle · qf-right-triangle-determination · candidate





## 36. Identifying a Right Triangle

A nondegenerate triangle has interior angles 35°, 80°, 65°. Is it a right triangle?

View answer and explanationNo

Although the angles sum to 180°, none is 90°, so the triangle is not right.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-right-triangle-determination-0006/)

Right triangle · qf-right-triangle-determination · candidate





## 37. Identifying a Right Triangle

A nondegenerate triangle has interior angles 40°, 90°, 50°. Is it a right triangle?

View answer and explanationYes

The angles sum to 180° and one angle is 90°, so the triangle is right.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-right-triangle-determination-0007/)

Right triangle · qf-right-triangle-determination · candidate





## 38. Identifying a Right Triangle

A nondegenerate triangle has interior angles 45°, 80°, 55°. Is it a right triangle?

View answer and explanationNo

Although the angles sum to 180°, none is 90°, so the triangle is not right.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-right-triangle-determination-0008/)

Right triangle · qf-right-triangle-determination · candidate





## 39. Identifying a Right Triangle

A nondegenerate triangle has interior angles 50°, 90°, 40°. Is it a right triangle?

View answer and explanationYes

The angles sum to 180° and one angle is 90°, so the triangle is right.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-right-triangle-determination-0009/)

Right triangle · qf-right-triangle-determination · candidate





## 40. Identifying a Right Triangle

A nondegenerate triangle has interior angles 55°, 80°, 45°. Is it a right triangle?

View answer and explanationNo

Although the angles sum to 180°, none is 90°, so the triangle is not right.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-right-triangle-determination-0010/)

Right triangle · qf-right-triangle-determination · candidate





## 41. Identifying a Right Triangle

A nondegenerate triangle has interior angles 60°, 90°, 30°. Is it a right triangle?

View answer and explanationYes

The angles sum to 180° and one angle is 90°, so the triangle is right.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-right-triangle-determination-0011/)

Right triangle · qf-right-triangle-determination · candidate





## 42. Identifying a Right Triangle

A nondegenerate triangle has interior angles 65°, 80°, 35°. Is it a right triangle?

View answer and explanationNo

Although the angles sum to 180°, none is 90°, so the triangle is not right.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-right-triangle-determination-0012/)

Right triangle · qf-right-triangle-determination · candidate





## 43. Identifying a Right Triangle

A nondegenerate triangle has interior angles 70°, 90°, 20°. Is it a right triangle?

View answer and explanationYes

The angles sum to 180° and one angle is 90°, so the triangle is right.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-right-triangle-determination-0013/)

Right triangle · qf-right-triangle-determination · candidate





## 44. Identifying a Right Triangle

A nondegenerate triangle has interior angles 75°, 80°, 25°. Is it a right triangle?

View answer and explanationNo

Although the angles sum to 180°, none is 90°, so the triangle is not right.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-right-triangle-determination-0014/)

Right triangle · qf-right-triangle-determination · candidate





## 45. Identifying a Right Triangle

A nondegenerate triangle has interior angles 80°, 90°, 10°. Is it a right triangle?

View answer and explanationYes

The angles sum to 180° and one angle is 90°, so the triangle is right.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-right-triangle-determination-0015/)

Right triangle · qf-right-triangle-determination · candidate





## 46. Identifying a Right Triangle

A nondegenerate triangle has interior angles 80°, 11°, 89°. Is it a right triangle?

View answer and explanationNo

Although the angles sum to 180°, none is 90°, so the triangle is not right.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-right-triangle-determination-0016/)

Right triangle · qf-right-triangle-determination · candidate





## 47. Identifying a Right Triangle

A nondegenerate triangle has interior angles 90°, 15°, 75°. Is it a right triangle?

View answer and explanationYes

The angles sum to 180° and one angle is 90°, so the triangle is right.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-right-triangle-determination-0017/)

Right triangle · qf-right-triangle-determination · candidate





## 48. Identifying a Right Triangle

A nondegenerate triangle has interior angles 80°, 19°, 81°. Is it a right triangle?

View answer and explanationNo

Although the angles sum to 180°, none is 90°, so the triangle is not right.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-right-triangle-determination-0018/)

Right triangle · qf-right-triangle-determination · candidate





## 49. Identifying a Right Triangle

A nondegenerate triangle has interior angles 90°, 23°, 67°. Is it a right triangle?

View answer and explanationYes

The angles sum to 180° and one angle is 90°, so the triangle is right.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-right-triangle-determination-0019/)

Right triangle · qf-right-triangle-determination · candidate





## 50. Identifying a Right Triangle

A nondegenerate triangle has interior angles 80°, 27°, 73°. Is it a right triangle?

View answer and explanationNo

Although the angles sum to 180°, none is 90°, so the triangle is not right.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-right-triangle-determination-0020/)

Right triangle · qf-right-triangle-determination · candidate





## 51. Identifying a Right Triangle

A nondegenerate triangle has interior angles 90°, 31°, 59°. Is it a right triangle?

View answer and explanationYes

The angles sum to 180° and one angle is 90°, so the triangle is right.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-right-triangle-determination-0021/)

Right triangle · qf-right-triangle-determination · candidate





## 52. Identifying a Right Triangle

A nondegenerate triangle has interior angles 80°, 35°, 65°. Is it a right triangle?

View answer and explanationNo

Although the angles sum to 180°, none is 90°, so the triangle is not right.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-right-triangle-determination-0022/)

Right triangle · qf-right-triangle-determination · candidate





## 53. Identifying a Right Triangle

A nondegenerate triangle has interior angles 90°, 39°, 51°. Is it a right triangle?

View answer and explanationYes

The angles sum to 180° and one angle is 90°, so the triangle is right.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-right-triangle-determination-0023/)

Right triangle · qf-right-triangle-determination · candidate





## 54. Identifying a Right Triangle

A nondegenerate triangle has interior angles 80°, 43°, 57°. Is it a right triangle?

View answer and explanationNo

Although the angles sum to 180°, none is 90°, so the triangle is not right.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-right-triangle-determination-0024/)

Right triangle · qf-right-triangle-determination · candidate





## 55. Identifying a Right Triangle

A nondegenerate triangle has interior angles 90°, 47°, 43°. Is it a right triangle?

View answer and explanationYes

The angles sum to 180° and one angle is 90°, so the triangle is right.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-right-triangle-determination-0025/)

Right triangle · qf-right-triangle-determination · candidate





## 56. Identifying a Right Triangle

A nondegenerate triangle has interior angles 80°, 51°, 49°. Is it a right triangle?

View answer and explanationNo

Although the angles sum to 180°, none is 90°, so the triangle is not right.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-right-triangle-determination-0026/)

Right triangle · qf-right-triangle-determination · candidate





## 57. Identifying a Right Triangle

A nondegenerate triangle has interior angles 90°, 55°, 35°. Is it a right triangle?

View answer and explanationYes

The angles sum to 180° and one angle is 90°, so the triangle is right.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-right-triangle-determination-0027/)

Right triangle · qf-right-triangle-determination · candidate





## 58. Identifying a Right Triangle

A nondegenerate triangle has interior angles 80°, 59°, 41°. Is it a right triangle?

View answer and explanationNo

Although the angles sum to 180°, none is 90°, so the triangle is not right.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-right-triangle-determination-0028/)

Right triangle · qf-right-triangle-determination · candidate





## 59. Identifying a Right Triangle

A nondegenerate triangle has interior angles 90°, 63°, 27°. Is it a right triangle?

View answer and explanationYes

The angles sum to 180° and one angle is 90°, so the triangle is right.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-right-triangle-determination-0029/)

Right triangle · qf-right-triangle-determination · candidate





## 60. Identifying a Right Triangle

A nondegenerate triangle has interior angles 80°, 67°, 33°. Is it a right triangle?

View answer and explanationNo

Although the angles sum to 180°, none is 90°, so the triangle is not right.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-right-triangle-determination-0030/)

Right triangle · qf-right-triangle-determination · candidate





## 61. Missing Side by the Pythagorean Theorem

A right triangle has leg lengths 3 and 4. Find the hypotenuse length c.

View answer and explanation5

c=√(3²+4²)=5.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-missing-side-0001/)

Pythagorean theorem · qf-pythagorean-missing-side · candidate





## 62. Missing Side by the Pythagorean Theorem

A right triangle has one leg of length 12 and a hypotenuse of length 13. Find the other leg length a.

View answer and explanation5

a=√(13²−12²)=5.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-missing-side-0002/)

Pythagorean theorem · qf-pythagorean-missing-side · candidate





## 63. Missing Side by the Pythagorean Theorem

A right triangle has leg lengths 8 and 15. Find the hypotenuse length c.

View answer and explanation17

c=√(8²+15²)=17.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-missing-side-0003/)

Pythagorean theorem · qf-pythagorean-missing-side · candidate





## 64. Missing Side by the Pythagorean Theorem

A right triangle has one leg of length 24 and a hypotenuse of length 25. Find the other leg length a.

View answer and explanation7

a=√(25²−24²)=7.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-missing-side-0004/)

Pythagorean theorem · qf-pythagorean-missing-side · candidate





## 65. Missing Side by the Pythagorean Theorem

A right triangle has leg lengths 20 and 21. Find the hypotenuse length c.

View answer and explanation29

c=√(20²+21²)=29.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-missing-side-0005/)

Pythagorean theorem · qf-pythagorean-missing-side · candidate





## 66. Missing Side by the Pythagorean Theorem

A right triangle has one leg of length 40 and a hypotenuse of length 41. Find the other leg length a.

View answer and explanation9

a=√(41²−40²)=9.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-missing-side-0006/)

Pythagorean theorem · qf-pythagorean-missing-side · candidate





## 67. Missing Side by the Pythagorean Theorem

A right triangle has leg lengths 12 and 35. Find the hypotenuse length c.

View answer and explanation37

c=√(12²+35²)=37.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-missing-side-0007/)

Pythagorean theorem · qf-pythagorean-missing-side · candidate





## 68. Missing Side by the Pythagorean Theorem

A right triangle has one leg of length 60 and a hypotenuse of length 61. Find the other leg length a.

View answer and explanation11

a=√(61²−60²)=11.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-missing-side-0008/)

Pythagorean theorem · qf-pythagorean-missing-side · candidate





## 69. Missing Side by the Pythagorean Theorem

A right triangle has leg lengths 28 and 45. Find the hypotenuse length c.

View answer and explanation53

c=√(28²+45²)=53.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-missing-side-0009/)

Pythagorean theorem · qf-pythagorean-missing-side · candidate





## 70. Missing Side by the Pythagorean Theorem

A right triangle has one leg of length 56 and a hypotenuse of length 65. Find the other leg length a.

View answer and explanation33

a=√(65²−56²)=33.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-missing-side-0010/)

Pythagorean theorem · qf-pythagorean-missing-side · candidate





## 71. Missing Side by the Pythagorean Theorem

A right triangle has leg lengths 16 and 63. Find the hypotenuse length c.

View answer and explanation65

c=√(16²+63²)=65.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-missing-side-0011/)

Pythagorean theorem · qf-pythagorean-missing-side · candidate





## 72. Missing Side by the Pythagorean Theorem

A right triangle has one leg of length 55 and a hypotenuse of length 73. Find the other leg length a.

View answer and explanation48

a=√(73²−55²)=48.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-missing-side-0012/)

Pythagorean theorem · qf-pythagorean-missing-side · candidate





## 73. Missing Side by the Pythagorean Theorem

A right triangle has leg lengths 13 and 84. Find the hypotenuse length c.

View answer and explanation85

c=√(13²+84²)=85.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-missing-side-0013/)

Pythagorean theorem · qf-pythagorean-missing-side · candidate





## 74. Missing Side by the Pythagorean Theorem

A right triangle has one leg of length 77 and a hypotenuse of length 85. Find the other leg length a.

View answer and explanation36

a=√(85²−77²)=36.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-missing-side-0014/)

Pythagorean theorem · qf-pythagorean-missing-side · candidate





## 75. Missing Side by the Pythagorean Theorem

A right triangle has leg lengths 39 and 80. Find the hypotenuse length c.

View answer and explanation89

c=√(39²+80²)=89.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-missing-side-0015/)

Pythagorean theorem · qf-pythagorean-missing-side · candidate





## 76. Missing Side by the Pythagorean Theorem

A right triangle has one leg of length 72 and a hypotenuse of length 97. Find the other leg length a.

View answer and explanation65

a=√(97²−72²)=65.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-missing-side-0016/)

Pythagorean theorem · qf-pythagorean-missing-side · candidate





## 77. Missing Side by the Pythagorean Theorem

A right triangle has leg lengths 20 and 99. Find the hypotenuse length c.

View answer and explanation101

c=√(20²+99²)=101.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-missing-side-0017/)

Pythagorean theorem · qf-pythagorean-missing-side · candidate





## 78. Missing Side by the Pythagorean Theorem

A right triangle has one leg of length 91 and a hypotenuse of length 109. Find the other leg length a.

View answer and explanation60

a=√(109²−91²)=60.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-missing-side-0018/)

Pythagorean theorem · qf-pythagorean-missing-side · candidate





## 79. Missing Side by the Pythagorean Theorem

A right triangle has leg lengths 15 and 112. Find the hypotenuse length c.

View answer and explanation113

c=√(15²+112²)=113.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-missing-side-0019/)

Pythagorean theorem · qf-pythagorean-missing-side · candidate





## 80. Missing Side by the Pythagorean Theorem

A right triangle has one leg of length 117 and a hypotenuse of length 125. Find the other leg length a.

View answer and explanation44

a=√(125²−117²)=44.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-missing-side-0020/)

Pythagorean theorem · qf-pythagorean-missing-side · candidate





## 81. Missing Side by the Pythagorean Theorem

A right triangle has leg lengths 6 and 8. Find the hypotenuse length c.

View answer and explanation10

c=√(6²+8²)=10.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-missing-side-0021/)

Pythagorean theorem · qf-pythagorean-missing-side · candidate





## 82. Missing Side by the Pythagorean Theorem

A right triangle has one leg of length 24 and a hypotenuse of length 26. Find the other leg length a.

View answer and explanation10

a=√(26²−24²)=10.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-missing-side-0022/)

Pythagorean theorem · qf-pythagorean-missing-side · candidate





## 83. Missing Side by the Pythagorean Theorem

A right triangle has leg lengths 16 and 30. Find the hypotenuse length c.

View answer and explanation34

c=√(16²+30²)=34.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-missing-side-0023/)

Pythagorean theorem · qf-pythagorean-missing-side · candidate





## 84. Missing Side by the Pythagorean Theorem

A right triangle has one leg of length 48 and a hypotenuse of length 50. Find the other leg length a.

View answer and explanation14

a=√(50²−48²)=14.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-missing-side-0024/)

Pythagorean theorem · qf-pythagorean-missing-side · candidate





## 85. Missing Side by the Pythagorean Theorem

A right triangle has leg lengths 40 and 42. Find the hypotenuse length c.

View answer and explanation58

c=√(40²+42²)=58.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-missing-side-0025/)

Pythagorean theorem · qf-pythagorean-missing-side · candidate





## 86. Counterexample to the Pythagorean Relation

Which positive triple does not satisfy a²+b²=c², taking the last number in each triple as c?
A: 3, 4, 7; B: 3, 4, 5; C: 3, 4, 5; D: 3, 4, 5

View answer and explanationA

For option A, a²+b² is not equal to c², so it is the counterexample to the stated squared relation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-counterexample-0001/)

Pythagorean theorem · qf-pythagorean-counterexample · candidate





## 87. Counterexample to the Pythagorean Relation

Which positive triple does not satisfy a²+b²=c², taking the last number in each triple as c?
A: 5, 12, 13; B: 5, 13, 13; C: 5, 12, 13; D: 5, 12, 13

View answer and explanationB

For option B, a²+b² is not equal to c², so it is the counterexample to the stated squared relation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-counterexample-0002/)

Pythagorean theorem · qf-pythagorean-counterexample · candidate





## 88. Counterexample to the Pythagorean Relation

Which positive triple does not satisfy a²+b²=c², taking the last number in each triple as c?
A: 8, 15, 17; B: 8, 15, 17; C: 10, 15, 17; D: 8, 15, 17

View answer and explanationC

For option C, a²+b² is not equal to c², so it is the counterexample to the stated squared relation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-counterexample-0003/)

Pythagorean theorem · qf-pythagorean-counterexample · candidate





## 89. Counterexample to the Pythagorean Relation

Which positive triple does not satisfy a²+b²=c², taking the last number in each triple as c?
A: 7, 24, 25; B: 7, 24, 25; C: 7, 24, 25; D: 7, 24, 27

View answer and explanationD

For option D, a²+b² is not equal to c², so it is the counterexample to the stated squared relation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-counterexample-0004/)

Pythagorean theorem · qf-pythagorean-counterexample · candidate





## 90. Counterexample to the Pythagorean Relation

Which positive triple does not satisfy a²+b²=c², taking the last number in each triple as c?
A: 20, 22, 29; B: 20, 21, 29; C: 20, 21, 29; D: 20, 21, 29

View answer and explanationA

For option A, a²+b² is not equal to c², so it is the counterexample to the stated squared relation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-counterexample-0005/)

Pythagorean theorem · qf-pythagorean-counterexample · candidate





## 91. Counterexample to the Pythagorean Relation

Which positive triple does not satisfy a²+b²=c², taking the last number in each triple as c?
A: 9, 40, 41; B: 11, 40, 41; C: 9, 40, 41; D: 9, 40, 41

View answer and explanationB

For option B, a²+b² is not equal to c², so it is the counterexample to the stated squared relation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-counterexample-0006/)

Pythagorean theorem · qf-pythagorean-counterexample · candidate





## 92. Counterexample to the Pythagorean Relation

Which positive triple does not satisfy a²+b²=c², taking the last number in each triple as c?
A: 12, 35, 37; B: 12, 35, 37; C: 12, 35, 39; D: 12, 35, 37

View answer and explanationC

For option C, a²+b² is not equal to c², so it is the counterexample to the stated squared relation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-counterexample-0007/)

Pythagorean theorem · qf-pythagorean-counterexample · candidate





## 93. Counterexample to the Pythagorean Relation

Which positive triple does not satisfy a²+b²=c², taking the last number in each triple as c?
A: 11, 60, 61; B: 11, 60, 61; C: 11, 60, 61; D: 11, 61, 61

View answer and explanationD

For option D, a²+b² is not equal to c², so it is the counterexample to the stated squared relation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-counterexample-0008/)

Pythagorean theorem · qf-pythagorean-counterexample · candidate





## 94. Counterexample to the Pythagorean Relation

Which positive triple does not satisfy a²+b²=c², taking the last number in each triple as c?
A: 30, 45, 53; B: 28, 45, 53; C: 28, 45, 53; D: 28, 45, 53

View answer and explanationA

For option A, a²+b² is not equal to c², so it is the counterexample to the stated squared relation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-counterexample-0009/)

Pythagorean theorem · qf-pythagorean-counterexample · candidate





## 95. Counterexample to the Pythagorean Relation

Which positive triple does not satisfy a²+b²=c², taking the last number in each triple as c?
A: 33, 56, 65; B: 33, 56, 67; C: 33, 56, 65; D: 33, 56, 65

View answer and explanationB

For option B, a²+b² is not equal to c², so it is the counterexample to the stated squared relation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-counterexample-0010/)

Pythagorean theorem · qf-pythagorean-counterexample · candidate





## 96. Counterexample to the Pythagorean Relation

Which positive triple does not satisfy a²+b²=c², taking the last number in each triple as c?
A: 16, 63, 65; B: 16, 63, 65; C: 16, 64, 65; D: 16, 63, 65

View answer and explanationC

For option C, a²+b² is not equal to c², so it is the counterexample to the stated squared relation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-counterexample-0011/)

Pythagorean theorem · qf-pythagorean-counterexample · candidate





## 97. Counterexample to the Pythagorean Relation

Which positive triple does not satisfy a²+b²=c², taking the last number in each triple as c?
A: 48, 55, 73; B: 48, 55, 73; C: 48, 55, 73; D: 50, 55, 73

View answer and explanationD

For option D, a²+b² is not equal to c², so it is the counterexample to the stated squared relation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-counterexample-0012/)

Pythagorean theorem · qf-pythagorean-counterexample · candidate





## 98. Counterexample to the Pythagorean Relation

Which positive triple does not satisfy a²+b²=c², taking the last number in each triple as c?
A: 13, 84, 87; B: 13, 84, 85; C: 13, 84, 85; D: 13, 84, 85

View answer and explanationA

For option A, a²+b² is not equal to c², so it is the counterexample to the stated squared relation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-counterexample-0013/)

Pythagorean theorem · qf-pythagorean-counterexample · candidate





## 99. Counterexample to the Pythagorean Relation

Which positive triple does not satisfy a²+b²=c², taking the last number in each triple as c?
A: 36, 77, 85; B: 36, 78, 85; C: 36, 77, 85; D: 36, 77, 85

View answer and explanationB

For option B, a²+b² is not equal to c², so it is the counterexample to the stated squared relation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-counterexample-0014/)

Pythagorean theorem · qf-pythagorean-counterexample · candidate





## 100. Counterexample to the Pythagorean Relation

Which positive triple does not satisfy a²+b²=c², taking the last number in each triple as c?
A: 39, 80, 89; B: 39, 80, 89; C: 41, 80, 89; D: 39, 80, 89

View answer and explanationC

For option C, a²+b² is not equal to c², so it is the counterexample to the stated squared relation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260912-pythagorean-counterexample-0015/)

Pythagorean theorem · qf-pythagorean-counterexample · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20260912-geometry-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20260912-geometry-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20260912-geometry-foundations/validation-report.json)