When using an image, transcribe any uncertain label yourself. Keep “given in the problem,” “marked in the diagram” and “looks that way” separate so an assumed property does not become a premise halfway through a proof.
A familiar side calculation can still be unjustified
Consider this original fictional exercise: triangle ABC has AB = 9 and BC = 12. Find AC. No angle or third-side information has been supplied.
Applying the Pythagorean theorem immediately gives sqrt(9² + 12²) = 15. The arithmetic is correct under an additional condition: angle ABC must be a right angle, making AB and BC the legs. The stated problem has not established that condition, so it does not determine AC uniquely.
OpenStax's triangle and Pythagorean-theorem section, checked October 24, 2026, states the relation for a right triangle. The side opposite the right angle is the hypotenuse. Identifying that side is part of choosing the equation.
Build a counterexample to the assumed angle
These coordinate constructions are authored checks, not measurements from a drawing or results of a ChatGPT session:
| Construction | Coordinates | What follows |
|---|---|---|
| Right angle at B | B(0,0), A(9,0), C(0,12) | AB = 9, BC = 12, AC = 15 |
| Different angle at B | B(0,0), A(9,0), C(6,6sqrt(3)) | AB = 9, BC = 12, AC = sqrt(117) |
For the second construction, BC² = 6² + (6sqrt(3))² = 36 + 108 = 144, so BC is still 12. But AC² = (6 - 9)² + (6sqrt(3))² = 9 + 108 = 117. Thus AC is about 10.82, not 15.
Both triangles satisfy the two stated side lengths. Their different third sides show why those lengths alone do not determine the answer. The coordinate distance calculations use perpendicular coordinate axes; they do not assume that the triangle's angle at B is right.
Ask for the missing premise explicitly
OpenAI's prompting guidance supports stating context and output requirements. In a geometry request, that means preserving the labels and asking for a reason before the calculation.
Review this Euclidean geometry problem using only the stated facts.
Triangle ABC has AB = 9 and BC = 12. We want AC.
No angle, parallel-line, equal-side or right-angle condition is given.
My attempt is AC² = 9² + 12², so AC = 15.
List the facts and identify the theorem my attempt uses. State its
required premises and whether the problem supplies them. Do not infer
a right angle from how a sketch looks. If the result is undetermined,
explain what additional information would suffice. Give two valid
constructions satisfying the known lengths but with different AC.
Then solve conditionally if angle ABC is explicitly 90 degrees.
Keep given facts, deductions and extra assumptions visibly separate.
If your actual problem contains a small square at B, transcribe it as angle ABC = 90°. The conditional result then becomes justified: AC = 15. If the right angle is instead at A, BC is the hypotenuse and the equation becomes 9² + AC² = 12², giving AC = sqrt(63). The location of the angle changes the calculation.
Review a proof by its reasons
Write a reason beside each step: a supplied fact, a definition or a theorem whose premises have already been established. Circle any step relying only on the picture's apparent symmetry or scale. Ask the model about that exact step rather than requesting a more convincing full proof.
The angle-comparison activity shows why rotation and displayed ray length do not change an angle's opening. Here, the task is to decide what the premises establish before computing. Finish with either a justified result or a precise missing-information statement, and keep the original labels beside your reasoning for the instructor's review.