Test a Right Angle from Three Sides
For a valid triangle, equality between the largest side’s square and the sum of the other two squares establishes a right angle opposite the largest side.
About 20 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: A valid triangle has complete side lengths (6, 8, 10). Which justified conclusion is correct? Candidate answers: (A) Cannot decide from complete side lengths (B) Not right: squared-side equality fails (C) Right triangle, because triangle inequality holds (D) Right triangle: squared-side equality holds.
- Independent check inputs: A valid triangle has complete side lengths (9, 12, 16). Which justified conclusion is correct? Candidate answers: (A) Cannot decide from complete side lengths (B) Not right: squared-side equality fails (C) Right triangle, because triangle inequality holds (D) Right triangle: squared-side equality holds.
Learn the idea
Dilations change lengths by a common factor and preserve shape. Right-triangle relationships connect squared perpendicular lengths; their converse tests whether supplied side lengths support a right angle, including in a three-dimensional diagonal model. For a valid triangle, equality between the largest side’s square and the sum of the other two squares establishes a right angle opposite the largest side. This is the theorem’s converse.
Here is the method on the supplied teaching case. The largest side is 10. Compare 6²+8²=100 with 10²=100. They agree, so the Pythagorean converse establishes that the triangle is right.
Use the largest side as the candidate hypotenuse. Triangle inequality alone establishes a possible triangle rather than a right triangle; a near-equality from rounded measurements needs separate error analysis.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: A valid triangle has complete side lengths (6, 8, 10). Which justified conclusion is correct? Candidate answers: (A) Cannot decide from complete side lengths (B) Not right: squared-side equality fails (C) Right triangle, because triangle inequality holds (D) Right triangle: squared-side equality holds. Solution: The largest side is 10. Compare 6²+8²=100 with 10²=100. They agree, so the Pythagorean converse establishes that the triangle is right. The final answer is Right triangle: squared-side equality holds. Use the largest side as the candidate hypotenuse. Triangle inequality alone establishes a possible triangle rather than a right triangle; a near-equality from rounded measurements needs separate error analysis.
Check your reasoning
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Helpful tips
- For a valid triangle, equality between the largest side’s square and the sum of the other two squares establishes a right angle opposite the largest side.
- Use the largest side as the candidate hypotenuse.
See the answer and explanation
Not right: squared-side equality fails
The largest side is 16. Compare 9²+12²=225 with 16²=256. They differ, so the Pythagorean converse establishes that the triangle is not right.
Take it further
Changed-input task: A valid triangle has complete side lengths (27, 36, 46). Which justified conclusion is correct? Candidate answers: (A) Cannot decide from complete side lengths (B) Not right: squared-side equality fails (C) Right triangle, because triangle inequality holds (D) Right triangle: squared-side equality holds. Show the mathematical reason, not only the selected answer. Expected result: Not right: squared-side equality fails. The largest side is 46. Compare 27²+36²=2025 with 46²=2116. They differ, so the Pythagorean converse establishes that the triangle is not right.
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