8.G.B.6
8th Grade
Geometry
Converse of Pythagorean Theorem
Explain a proof of the Pythagorean Theorem and its converse.
How to explain it
The anchor students hold onto: Sort sides: a ≤ b ≤ c. Compute a²+b² and compare to c². Equal → right. Greater → acute. Less → obtuse.
The converse underpins Pythagorean Distance on the Coordinate Plane and applies whenever right-angle verification appears in geometry proofs.
Worked examples
Example 1
Explain the proof of a²+b²=c².
Step 1Each side forms a square with area a², b², or c².
Step 2Leg-squares sum equals hyp-square: a²+b²=c².
AnswerThe area of the two leg-squares always equals the hypotenuse-square.
Example 2
Sides 6, 8, 10: right triangle?
Step 1Identify longest side: c = 10.
Step 2a²+b² = 6²+8² = 36+64 = 100 = 10².
Step 3100 = 100 → right triangle.
AnswerYes — 6, 8, 10 is a right triangle (6²+8²=10²).
Example 3
Classify: sides 5, 7, 9.
Step 1Longest side: c = 9.
Step 2a²+b² = 5²+7² = 25+49 = 74; c² = 81.
Step 374 < 81 → a²+b² < c² → obtuse.
AnswerObtuse triangle (a²+b² < c²).
Common mistakes
What students write
Using a²+b²=c² with any two sides — c must always be the LONGEST side.
The fix
Sort the three sides first: a ≤ b ≤ c. Use only the two shorter sides as a and b.
What students write
Confusing theorem and converse: the theorem proves right triangles; the converse tests for one.
The fix
Converse: if a²+b²=c², then right. If not equal, compare > or < c² to classify.
Teacher tip
Head off the two predictable errors before they happen. First: Sort the three sides first: a ≤ b ≤ c. Use only the two shorter sides as a and b. Second: Converse: if a²+b²=c², then right. If not equal, compare > or < c² to classify.