Do sides 9, 12 and 15 form a right triangle?
- 9 + 12 > 15, so a triangle is possible.
- 9² + 12² = 81 + 144 = 225 = 15².
- Yes: it is a scaled 3–4–5 right triangle.
Right triangles & Pythagorean triples
Test the longest side, rather than assuming a right angle.
The converse tests whether a triangle is right
Use the longest side and the converse to decide whether a triangle is right-angled.
For three positive side lengths, first check that the two shorter lengths sum to more than the longest; otherwise they do not form a non-degenerate triangle. Call the longest side c. If a² + b² = c², the triangle is right-angled, with the right angle opposite c.
Explore the relationship
Predict first. Change one value, then explain the result. Diagrams show the entered dimensions; displayed decimals are rounded.
Triples are integer side-length relationships
The positive integers 3, 4, 5 form a Pythagorean triple because 9 + 16 = 25. So do 5, 12, 13 and 8, 15, 17. Multiplying all three sides by the same positive factor preserves the relationship: 6, 8, 10 is a scaled 3–4–5 triangle.
Exact equality is different from rounded evidence
The lengths 6, 8 and 10 give exact equality. Measurements rounded to the nearest centimetre do not prove an angle is exactly 90°. Explain the precision of the measurements. Also, an equation with the wrong side chosen as c does not test the converse correctly.
TED-EdWatch · pause · explain
How many ways are there to prove the Pythagorean theorem? — Betty Fei
Choose one visual argument. Explain how it connects areas without relying only on a numerical example.
Open on YouTube →Worked examples
Test sides 5, 6 and 8
- 5 + 6 > 8, so these can form a triangle.
- 5² + 6² = 61, while 8² = 64.
- The triangle is not right-angled.
Guided Practice
Test sides 7, 24 and 25.
A first hint
Take 25 as the candidate hypotenuse.
Compare your reasoning
7 + 24 > 25 and 49 + 576 = 625 = 25². It is a right triangle.
Practice
Fresh questions · useful feedback
Build fluency, one decision at a time.
Work through eight questions. Two independent correct answers introduce a harder version; supported answers bring the next question back to the foundation. You can use hints at any time.
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Connections
Why does 1, 2, √5 satisfy the squared equation, while 1, 2, 3 does not even form a non-degenerate triangle?
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