| Term | Definition |
|---|---|
| Right triangle | A triangle with one angle equal to \(90^\circ\). |
| Legs | The two sides that form the right angle. |
| Hypotenuse | The side opposite the right angle; the longest side in a right triangle. |
| Pythagorean Theorem | The relationship \(a^2+b^2=c^2\) for right triangles. |
The theorem is associated with Pythagoras, but right-triangle relationships appeared in earlier Babylonian, Egyptian, and Indian mathematics. In Euclidean geometry, the theorem became a central tool for proof, measurement, and construction.
Build a large square of side length \(a+b\). Place four identical right triangles (legs \(a,b\), hypotenuse \(c\)) inside it so the uncovered center is a square of side \(c\).
Area of big square: \((a+b)^2\). Area of the same figure as pieces: \(4\left(\frac{1}{2}ab\right)+c^2=2ab+c^2\). Set them equal:
\[ (a+b)^2 = 2ab + c^2 \]
\[ a^2 + 2ab + b^2 = 2ab + c^2 \]
\[ a^2 + b^2 = c^2 \]
Answer: \(c=\sqrt{3^2+4^2}=\sqrt{25}=5\).
Answer: \(c=\sqrt{6^2+8^2}=\sqrt{100}=10\).
Answer: \(c=\sqrt{5^2+12^2}=\sqrt{169}=13\).
Answer: \(d=\sqrt{6^2+6^2}=\sqrt{72}=6\sqrt{2}\).
Answer: \(d=\sqrt{9^2+9^2}=\sqrt{162}=9\sqrt{2}\).
Answer: \(\sqrt{8^2-4^2}=\sqrt{64-16}=\sqrt{48}=4\sqrt{3}\).
Answer: \(\sqrt{15^2-10^2}=\sqrt{225-100}=\sqrt{125}=5\sqrt{5}\).
Answer: \(c=\sqrt{x^2+y^2}\).
Answer: They are two area expressions for the same large square: one as a full square of side \(a+b\), and one as four right triangles plus the center square of side \(c\).