Discover, geometrically prove, and apply the Pythagorean Theorem.
Imagine you are helping an architect build a scale model of a sculpture. One triangular frame is a right triangle with legs 9 ft and 12 ft. How long is the third side?
In any right triangle with legs \(a\) and \(b\), and hypotenuse \(c\):
\[ a^2 + b^2 = c^2 \]
The hypotenuse is opposite the right angle, so it is always the longest side.
We usually name vertices with uppercase letters and opposite sides with matching lowercase letters.
Example. If \(a=9\) and \(b=12\), find \(c\).
\[ \begin{aligned} c &= \sqrt{9^2 + 12^2} \\ &= \sqrt{81 + 144} \\ &= \sqrt{225} \\ &= 15 \end{aligned} \]
The third side is 15 ft.
Example. If \(c=13\) and \(a=5\), find \(b\).
\[ \begin{aligned} b &= \sqrt{c^2 - a^2} \\ &= \sqrt{13^2 - 5^2} \\ &= \sqrt{169 - 25} \\ &= \sqrt{144} \\ &= 12 \end{aligned} \]
Example. If \(c=10\) and \(a=2\), find \(b\).
\[ \begin{aligned} b &= \sqrt{10^2 - 2^2} \\ &= \sqrt{100 - 4} \\ &= \sqrt{96} \\ &= \sqrt{16 \cdot 6} \\ &= 4\sqrt{6} \end{aligned} \]
When solving \(c^2=289\), the equation has two numerical roots (\(\pm 17\)), but side lengths in geometry are nonnegative. So the triangle side is \(17\), not \(-17\).
If three side lengths satisfy \(a^2+b^2=c^2\), then the triangle is right-angled.
Example: \(7,24,25\) because \(7^2+24^2=49+576=625=25^2\).
Distance between points \(A(x_1,y_1)\) and \(B(x_2,y_2)\):
\[ AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \]
This is Pythagoras applied to horizontal and vertical change.
| 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 \(c\).