Invariants Math Example 4
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Example 4
mediumShow that for any right triangle with legs \(a, b\) and hypotenuse \(c\), the quantity \(a^2 + b^2 - c^2 = 0\) is an invariant (Pythagorean theorem).
Solution
- 1 The Pythagorean theorem states: for any right triangle, \(a^2 + b^2 = c^2\).
- 2 Rearranging: \(a^2 + b^2 - c^2 = 0\).
- 3 This holds for ALL right triangles, regardless of their size.
- 4 Example: \(a=3, b=4, c=5\): \(9+16-25=0\) ✓. \(a=5,b=12,c=13\): \(25+144-169=0\) ✓.
Answer
\(a^2 + b^2 - c^2 = 0\) for all right triangles
The Pythagorean relationship is an invariant of right triangles — it holds for every right triangle, making it a universal structural property.
About Invariants
Quantities or properties that remain unchanged during a process, operation, or transformation—values that stay the same no matter how the system is rearranged or acted upon.
Learn more about Invariants →More Invariants Examples
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