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.
In the Euclidean plane, P=(-23, -26) and Q=(-20, -22). Find the distance d(P,Q).
5
The coordinate difference is (3, 4), so d=√((3)²+(4)²)=5.
Reviewed step-by-step derivation
Euclidean length · qf-distance-coordinate · candidate
In the Euclidean plane, P=(-16, -15) and Q=(-21, -3). Find the distance d(P,Q).
13
The coordinate difference is (-5, 12), so d=√((-5)²+(12)²)=13.
Reviewed step-by-step derivation
Euclidean length · qf-distance-coordinate · candidate
In the Euclidean plane, P=(-9, -4) and Q=(-1, -19). Find the distance d(P,Q).
17
The coordinate difference is (8, -15), so d=√((8)²+(-15)²)=17.
Reviewed step-by-step derivation
Euclidean length · qf-distance-coordinate · candidate
In the Euclidean plane, P=(-2, 7) and Q=(-9, -17). Find the distance d(P,Q).
25
The coordinate difference is (-7, -24), so d=√((-7)²+(-24)²)=25.
Reviewed step-by-step derivation
Euclidean length · qf-distance-coordinate · candidate
In the Euclidean plane, P=(5, 18) and Q=(25, 39). Find the distance d(P,Q).
29
The coordinate difference is (20, 21), so d=√((20)²+(21)²)=29.
Reviewed step-by-step derivation
Euclidean length · qf-distance-coordinate · candidate
In the Euclidean plane, P=(12, -24) and Q=(3, 16). Find the distance d(P,Q).
41
The coordinate difference is (-9, 40), so d=√((-9)²+(40)²)=41.
Reviewed step-by-step derivation
Euclidean length · qf-distance-coordinate · candidate
In the Euclidean plane, P=(19, -13) and Q=(31, -48). Find the distance d(P,Q).
37
The coordinate difference is (12, -35), so d=√((12)²+(-35)²)=37.
Reviewed step-by-step derivation
Euclidean length · qf-distance-coordinate · candidate
In the Euclidean plane, P=(-21, -2) and Q=(-32, -62). Find the distance d(P,Q).
61
The coordinate difference is (-11, -60), so d=√((-11)²+(-60)²)=61.
Reviewed step-by-step derivation
Euclidean length · qf-distance-coordinate · candidate
In the Euclidean plane, P=(-14, 9) and Q=(14, 54). Find the distance d(P,Q).
53
The coordinate difference is (28, 45), so d=√((28)²+(45)²)=53.
Reviewed step-by-step derivation
Euclidean length · qf-distance-coordinate · candidate
In the Euclidean plane, P=(-7, 20) and Q=(-40, 76). Find the distance d(P,Q).
65
The coordinate difference is (-33, 56), so d=√((-33)²+(56)²)=65.
Reviewed step-by-step derivation
Euclidean length · qf-distance-coordinate · candidate
In the Euclidean plane, P=(0, -22) and Q=(16, -85). Find the distance d(P,Q).
65
The coordinate difference is (16, -63), so d=√((16)²+(-63)²)=65.
Reviewed step-by-step derivation
Euclidean length · qf-distance-coordinate · candidate
In the Euclidean plane, P=(7, -11) and Q=(-41, -66). Find the distance d(P,Q).
73
The coordinate difference is (-48, -55), so d=√((-48)²+(-55)²)=73.
Reviewed step-by-step derivation
Euclidean length · qf-distance-coordinate · candidate
In the Euclidean plane, P=(14, 0) and Q=(27, 84). Find the distance d(P,Q).
85
The coordinate difference is (13, 84), so d=√((13)²+(84)²)=85.
Reviewed step-by-step derivation
Euclidean length · qf-distance-coordinate · candidate
In the Euclidean plane, P=(21, 11) and Q=(-15, 88). Find the distance d(P,Q).
85
The coordinate difference is (-36, 77), so d=√((-36)²+(77)²)=85.
Reviewed step-by-step derivation
Euclidean length · qf-distance-coordinate · candidate
In the Euclidean plane, P=(-19, 22) and Q=(20, -58). Find the distance d(P,Q).
89
The coordinate difference is (39, -80), so d=√((39)²+(-80)²)=89.
Reviewed step-by-step derivation
Euclidean length · qf-distance-coordinate · candidate
In the Euclidean plane, P=(-12, -20) and Q=(-77, -92). Find the distance d(P,Q).
97
The coordinate difference is (-65, -72), so d=√((-65)²+(-72)²)=97.
Reviewed step-by-step derivation
Euclidean length · qf-distance-coordinate · candidate
In the Euclidean plane, P=(-5, -9) and Q=(15, 90). Find the distance d(P,Q).
101
The coordinate difference is (20, 99), so d=√((20)²+(99)²)=101.
Reviewed step-by-step derivation
Euclidean length · qf-distance-coordinate · candidate
In the Euclidean plane, P=(2, 2) and Q=(-58, 93). Find the distance d(P,Q).
109
The coordinate difference is (-60, 91), so d=√((-60)²+(91)²)=109.
Reviewed step-by-step derivation
Euclidean length · qf-distance-coordinate · candidate
In the Euclidean plane, P=(9, 13) and Q=(24, -99). Find the distance d(P,Q).
113
The coordinate difference is (15, -112), so d=√((15)²+(-112)²)=113.
Reviewed step-by-step derivation
Euclidean length · qf-distance-coordinate · candidate
In the Euclidean plane, P=(16, 24) and Q=(-28, -93). Find the distance d(P,Q).
125
The coordinate difference is (-44, -117), so d=√((-44)²+(-117)²)=125.
Reviewed step-by-step derivation
Euclidean length · qf-distance-coordinate · candidate
In the Euclidean plane, P=(23, -18) and Q=(29, -10). Find the distance d(P,Q).
10
The coordinate difference is (6, 8), so d=√((6)²+(8)²)=10.
Reviewed step-by-step derivation
Euclidean length · qf-distance-coordinate · candidate
In the Euclidean plane, P=(-17, -7) and Q=(-27, 17). Find the distance d(P,Q).
26
The coordinate difference is (-10, 24), so d=√((-10)²+(24)²)=26.
Reviewed step-by-step derivation
Euclidean length · qf-distance-coordinate · candidate
In the Euclidean plane, P=(-10, 4) and Q=(6, -26). Find the distance d(P,Q).
34
The coordinate difference is (16, -30), so d=√((16)²+(-30)²)=34.
Reviewed step-by-step derivation
Euclidean length · qf-distance-coordinate · candidate
In the Euclidean plane, P=(-3, 15) and Q=(-17, -33). Find the distance d(P,Q).
50
The coordinate difference is (-14, -48), so d=√((-14)²+(-48)²)=50.
Reviewed step-by-step derivation
Euclidean length · qf-distance-coordinate · candidate
In the Euclidean plane, P=(4, 26) and Q=(44, 68). Find the distance d(P,Q).
58
The coordinate difference is (40, 42), so d=√((40)²+(42)²)=58.
Reviewed step-by-step derivation
Euclidean length · qf-distance-coordinate · candidate
In the Euclidean plane, P=(11, -16) and Q=(-7, 64). Find the distance d(P,Q).
82
The coordinate difference is (-18, 80), so d=√((-18)²+(80)²)=82.
Reviewed step-by-step derivation
Euclidean length · qf-distance-coordinate · candidate
In the Euclidean plane, P=(18, -5) and Q=(42, -75). Find the distance d(P,Q).
74
The coordinate difference is (24, -70), so d=√((24)²+(-70)²)=74.
Reviewed step-by-step derivation
Euclidean length · qf-distance-coordinate · candidate
In the Euclidean plane, P=(-22, 6) and Q=(-44, -114). Find the distance d(P,Q).
122
The coordinate difference is (-22, -120), so d=√((-22)²+(-120)²)=122.
Reviewed step-by-step derivation
Euclidean length · qf-distance-coordinate · candidate
In the Euclidean plane, P=(-15, 17) and Q=(41, 107). Find the distance d(P,Q).
106
The coordinate difference is (56, 90), so d=√((56)²+(90)²)=106.
Reviewed step-by-step derivation
Euclidean length · qf-distance-coordinate · candidate
In the Euclidean plane, P=(-8, -25) and Q=(-74, 87). Find the distance d(P,Q).
130
The coordinate difference is (-66, 112), so d=√((-66)²+(112)²)=130.
Reviewed step-by-step derivation
Euclidean length · qf-distance-coordinate · candidate
A nondegenerate triangle has interior angles 10°, 90°, 80°. Is it a right triangle?
Yes
The angles sum to 180° and one angle is 90°, so the triangle is right.
Reviewed step-by-step derivation
Right triangle · qf-right-triangle-determination · candidate
A nondegenerate triangle has interior angles 15°, 80°, 85°. Is it a right triangle?
No
Although the angles sum to 180°, none is 90°, so the triangle is not right.
Reviewed step-by-step derivation
Right triangle · qf-right-triangle-determination · candidate
A nondegenerate triangle has interior angles 20°, 90°, 70°. Is it a right triangle?
Yes
The angles sum to 180° and one angle is 90°, so the triangle is right.
Reviewed step-by-step derivation
Right triangle · qf-right-triangle-determination · candidate
A nondegenerate triangle has interior angles 25°, 80°, 75°. Is it a right triangle?
No
Although the angles sum to 180°, none is 90°, so the triangle is not right.
Reviewed step-by-step derivation
Right triangle · qf-right-triangle-determination · candidate
A nondegenerate triangle has interior angles 30°, 90°, 60°. Is it a right triangle?
Yes
The angles sum to 180° and one angle is 90°, so the triangle is right.
Reviewed step-by-step derivation
Right triangle · qf-right-triangle-determination · candidate
A nondegenerate triangle has interior angles 35°, 80°, 65°. Is it a right triangle?
No
Although the angles sum to 180°, none is 90°, so the triangle is not right.
Reviewed step-by-step derivation
Right triangle · qf-right-triangle-determination · candidate
A nondegenerate triangle has interior angles 40°, 90°, 50°. Is it a right triangle?
Yes
The angles sum to 180° and one angle is 90°, so the triangle is right.
Reviewed step-by-step derivation
Right triangle · qf-right-triangle-determination · candidate
A nondegenerate triangle has interior angles 45°, 80°, 55°. Is it a right triangle?
No
Although the angles sum to 180°, none is 90°, so the triangle is not right.
Reviewed step-by-step derivation
Right triangle · qf-right-triangle-determination · candidate
A nondegenerate triangle has interior angles 50°, 90°, 40°. Is it a right triangle?
Yes
The angles sum to 180° and one angle is 90°, so the triangle is right.
Reviewed step-by-step derivation
Right triangle · qf-right-triangle-determination · candidate
A nondegenerate triangle has interior angles 55°, 80°, 45°. Is it a right triangle?
No
Although the angles sum to 180°, none is 90°, so the triangle is not right.
Reviewed step-by-step derivation
Right triangle · qf-right-triangle-determination · candidate
A nondegenerate triangle has interior angles 60°, 90°, 30°. Is it a right triangle?
Yes
The angles sum to 180° and one angle is 90°, so the triangle is right.
Reviewed step-by-step derivation
Right triangle · qf-right-triangle-determination · candidate
A nondegenerate triangle has interior angles 65°, 80°, 35°. Is it a right triangle?
No
Although the angles sum to 180°, none is 90°, so the triangle is not right.
Reviewed step-by-step derivation
Right triangle · qf-right-triangle-determination · candidate
A nondegenerate triangle has interior angles 70°, 90°, 20°. Is it a right triangle?
Yes
The angles sum to 180° and one angle is 90°, so the triangle is right.
Reviewed step-by-step derivation
Right triangle · qf-right-triangle-determination · candidate
A nondegenerate triangle has interior angles 75°, 80°, 25°. Is it a right triangle?
No
Although the angles sum to 180°, none is 90°, so the triangle is not right.
Reviewed step-by-step derivation
Right triangle · qf-right-triangle-determination · candidate
A nondegenerate triangle has interior angles 80°, 90°, 10°. Is it a right triangle?
Yes
The angles sum to 180° and one angle is 90°, so the triangle is right.
Reviewed step-by-step derivation
Right triangle · qf-right-triangle-determination · candidate
A nondegenerate triangle has interior angles 80°, 11°, 89°. Is it a right triangle?
No
Although the angles sum to 180°, none is 90°, so the triangle is not right.
Reviewed step-by-step derivation
Right triangle · qf-right-triangle-determination · candidate
A nondegenerate triangle has interior angles 90°, 15°, 75°. Is it a right triangle?
Yes
The angles sum to 180° and one angle is 90°, so the triangle is right.
Reviewed step-by-step derivation
Right triangle · qf-right-triangle-determination · candidate
A nondegenerate triangle has interior angles 80°, 19°, 81°. Is it a right triangle?
No
Although the angles sum to 180°, none is 90°, so the triangle is not right.
Reviewed step-by-step derivation
Right triangle · qf-right-triangle-determination · candidate
A nondegenerate triangle has interior angles 90°, 23°, 67°. Is it a right triangle?
Yes
The angles sum to 180° and one angle is 90°, so the triangle is right.
Reviewed step-by-step derivation
Right triangle · qf-right-triangle-determination · candidate
A nondegenerate triangle has interior angles 80°, 27°, 73°. Is it a right triangle?
No
Although the angles sum to 180°, none is 90°, so the triangle is not right.
Reviewed step-by-step derivation
Right triangle · qf-right-triangle-determination · candidate
A nondegenerate triangle has interior angles 90°, 31°, 59°. Is it a right triangle?
Yes
The angles sum to 180° and one angle is 90°, so the triangle is right.
Reviewed step-by-step derivation
Right triangle · qf-right-triangle-determination · candidate
A nondegenerate triangle has interior angles 80°, 35°, 65°. Is it a right triangle?
No
Although the angles sum to 180°, none is 90°, so the triangle is not right.
Reviewed step-by-step derivation
Right triangle · qf-right-triangle-determination · candidate
A nondegenerate triangle has interior angles 90°, 39°, 51°. Is it a right triangle?
Yes
The angles sum to 180° and one angle is 90°, so the triangle is right.
Reviewed step-by-step derivation
Right triangle · qf-right-triangle-determination · candidate
A nondegenerate triangle has interior angles 80°, 43°, 57°. Is it a right triangle?
No
Although the angles sum to 180°, none is 90°, so the triangle is not right.
Reviewed step-by-step derivation
Right triangle · qf-right-triangle-determination · candidate
A nondegenerate triangle has interior angles 90°, 47°, 43°. Is it a right triangle?
Yes
The angles sum to 180° and one angle is 90°, so the triangle is right.
Reviewed step-by-step derivation
Right triangle · qf-right-triangle-determination · candidate
A nondegenerate triangle has interior angles 80°, 51°, 49°. Is it a right triangle?
No
Although the angles sum to 180°, none is 90°, so the triangle is not right.
Reviewed step-by-step derivation
Right triangle · qf-right-triangle-determination · candidate
A nondegenerate triangle has interior angles 90°, 55°, 35°. Is it a right triangle?
Yes
The angles sum to 180° and one angle is 90°, so the triangle is right.
Reviewed step-by-step derivation
Right triangle · qf-right-triangle-determination · candidate
A nondegenerate triangle has interior angles 80°, 59°, 41°. Is it a right triangle?
No
Although the angles sum to 180°, none is 90°, so the triangle is not right.
Reviewed step-by-step derivation
Right triangle · qf-right-triangle-determination · candidate
A nondegenerate triangle has interior angles 90°, 63°, 27°. Is it a right triangle?
Yes
The angles sum to 180° and one angle is 90°, so the triangle is right.
Reviewed step-by-step derivation
Right triangle · qf-right-triangle-determination · candidate
A nondegenerate triangle has interior angles 80°, 67°, 33°. Is it a right triangle?
No
Although the angles sum to 180°, none is 90°, so the triangle is not right.
Reviewed step-by-step derivation
Right triangle · qf-right-triangle-determination · candidate
A right triangle has leg lengths 3 and 4. Find the hypotenuse length c.
5
c=√(3²+4²)=5.
Reviewed step-by-step derivation
Pythagorean theorem · qf-pythagorean-missing-side · candidate
A right triangle has one leg of length 12 and a hypotenuse of length 13. Find the other leg length a.
5
a=√(13²−12²)=5.
Reviewed step-by-step derivation
Pythagorean theorem · qf-pythagorean-missing-side · candidate
A right triangle has leg lengths 8 and 15. Find the hypotenuse length c.
17
c=√(8²+15²)=17.
Reviewed step-by-step derivation
Pythagorean theorem · qf-pythagorean-missing-side · candidate
A right triangle has one leg of length 24 and a hypotenuse of length 25. Find the other leg length a.
7
a=√(25²−24²)=7.
Reviewed step-by-step derivation
Pythagorean theorem · qf-pythagorean-missing-side · candidate
A right triangle has leg lengths 20 and 21. Find the hypotenuse length c.
29
c=√(20²+21²)=29.
Reviewed step-by-step derivation
Pythagorean theorem · qf-pythagorean-missing-side · candidate
A right triangle has one leg of length 40 and a hypotenuse of length 41. Find the other leg length a.
9
a=√(41²−40²)=9.
Reviewed step-by-step derivation
Pythagorean theorem · qf-pythagorean-missing-side · candidate
A right triangle has leg lengths 12 and 35. Find the hypotenuse length c.
37
c=√(12²+35²)=37.
Reviewed step-by-step derivation
Pythagorean theorem · qf-pythagorean-missing-side · candidate
A right triangle has one leg of length 60 and a hypotenuse of length 61. Find the other leg length a.
11
a=√(61²−60²)=11.
Reviewed step-by-step derivation
Pythagorean theorem · qf-pythagorean-missing-side · candidate
A right triangle has leg lengths 28 and 45. Find the hypotenuse length c.
53
c=√(28²+45²)=53.
Reviewed step-by-step derivation
Pythagorean theorem · qf-pythagorean-missing-side · candidate
A right triangle has one leg of length 56 and a hypotenuse of length 65. Find the other leg length a.
33
a=√(65²−56²)=33.
Reviewed step-by-step derivation
Pythagorean theorem · qf-pythagorean-missing-side · candidate
A right triangle has leg lengths 16 and 63. Find the hypotenuse length c.
65
c=√(16²+63²)=65.
Reviewed step-by-step derivation
Pythagorean theorem · qf-pythagorean-missing-side · candidate
A right triangle has one leg of length 55 and a hypotenuse of length 73. Find the other leg length a.
48
a=√(73²−55²)=48.
Reviewed step-by-step derivation
Pythagorean theorem · qf-pythagorean-missing-side · candidate
A right triangle has leg lengths 13 and 84. Find the hypotenuse length c.
85
c=√(13²+84²)=85.
Reviewed step-by-step derivation
Pythagorean theorem · qf-pythagorean-missing-side · candidate
A right triangle has one leg of length 77 and a hypotenuse of length 85. Find the other leg length a.
36
a=√(85²−77²)=36.
Reviewed step-by-step derivation
Pythagorean theorem · qf-pythagorean-missing-side · candidate
A right triangle has leg lengths 39 and 80. Find the hypotenuse length c.
89
c=√(39²+80²)=89.
Reviewed step-by-step derivation
Pythagorean theorem · qf-pythagorean-missing-side · candidate
A right triangle has one leg of length 72 and a hypotenuse of length 97. Find the other leg length a.
65
a=√(97²−72²)=65.
Reviewed step-by-step derivation
Pythagorean theorem · qf-pythagorean-missing-side · candidate
A right triangle has leg lengths 20 and 99. Find the hypotenuse length c.
101
c=√(20²+99²)=101.
Reviewed step-by-step derivation
Pythagorean theorem · qf-pythagorean-missing-side · candidate
A right triangle has one leg of length 91 and a hypotenuse of length 109. Find the other leg length a.
60
a=√(109²−91²)=60.
Reviewed step-by-step derivation
Pythagorean theorem · qf-pythagorean-missing-side · candidate
A right triangle has leg lengths 15 and 112. Find the hypotenuse length c.
113
c=√(15²+112²)=113.
Reviewed step-by-step derivation
Pythagorean theorem · qf-pythagorean-missing-side · candidate
A right triangle has one leg of length 117 and a hypotenuse of length 125. Find the other leg length a.
44
a=√(125²−117²)=44.
Reviewed step-by-step derivation
Pythagorean theorem · qf-pythagorean-missing-side · candidate
A right triangle has leg lengths 6 and 8. Find the hypotenuse length c.
10
c=√(6²+8²)=10.
Reviewed step-by-step derivation
Pythagorean theorem · qf-pythagorean-missing-side · candidate
A right triangle has one leg of length 24 and a hypotenuse of length 26. Find the other leg length a.
10
a=√(26²−24²)=10.
Reviewed step-by-step derivation
Pythagorean theorem · qf-pythagorean-missing-side · candidate
A right triangle has leg lengths 16 and 30. Find the hypotenuse length c.
34
c=√(16²+30²)=34.
Reviewed step-by-step derivation
Pythagorean theorem · qf-pythagorean-missing-side · candidate
A right triangle has one leg of length 48 and a hypotenuse of length 50. Find the other leg length a.
14
a=√(50²−48²)=14.
Reviewed step-by-step derivation
Pythagorean theorem · qf-pythagorean-missing-side · candidate
A right triangle has leg lengths 40 and 42. Find the hypotenuse length c.
58
c=√(40²+42²)=58.
Reviewed step-by-step derivation
Pythagorean theorem · qf-pythagorean-missing-side · candidate
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
A
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
Pythagorean theorem · qf-pythagorean-counterexample · candidate
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
B
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
Pythagorean theorem · qf-pythagorean-counterexample · candidate
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
C
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
Pythagorean theorem · qf-pythagorean-counterexample · candidate
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
D
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
Pythagorean theorem · qf-pythagorean-counterexample · candidate
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
A
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
Pythagorean theorem · qf-pythagorean-counterexample · candidate
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
B
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
Pythagorean theorem · qf-pythagorean-counterexample · candidate
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
C
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
Pythagorean theorem · qf-pythagorean-counterexample · candidate
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
D
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
Pythagorean theorem · qf-pythagorean-counterexample · candidate
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
A
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
Pythagorean theorem · qf-pythagorean-counterexample · candidate
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
B
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
Pythagorean theorem · qf-pythagorean-counterexample · candidate
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
C
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
Pythagorean theorem · qf-pythagorean-counterexample · candidate
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
D
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
Pythagorean theorem · qf-pythagorean-counterexample · candidate
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
A
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
Pythagorean theorem · qf-pythagorean-counterexample · candidate
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
B
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
Pythagorean theorem · qf-pythagorean-counterexample · candidate
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
C
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
Pythagorean theorem · qf-pythagorean-counterexample · candidate
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