Geometry and Trigonometry - Pythagorean theorem and its applications

The Right-Angled Triangle: The Pythagorean theorem exclusively applies to right-angled triangles, which are triangles containing one $90^{\circ}$ angle. The side opposite this right angle is called the hypotenuse, and it is always the longest side of the triangle. Visually, if you look at a right triangle, the hypotenuse is the diagonal side that does not touch the square symbol representing the $90^{\circ}$ angle.

The Fundamental Theorem: The theorem states that in any right-angled triangle, the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares whose sides are the two legs (the shorter sides). This is represented by the equation $a^2 + b^2 = c^2$, where $c$ is the hypotenuse.

Geometric Visualization: Imagine a right triangle where a physical square is drawn outwards from each of the three sides. If the sides are $3$, $4$, and $5$, the areas of the squares on the legs are $9$ ($3 \times 3$) and $16$ ($4 \times 4$). The area of the square on the hypotenuse is $25$ ($5 \times 5$). Note that $9 + 16 = 25$, visually confirming the theorem.

Finding the Hypotenuse: To find the length of the hypotenuse ($c$) when the lengths of the legs ($a$ and $b$) are known, you must sum the squares of the legs and then take the square root of the result: $c = \sqrt{a^2 + b^2}$.

Finding a Shorter Side: To find a missing leg ($a$ or $b$) when the hypotenuse ($c$) and one leg are known, subtract the square of the known leg from the square of the hypotenuse and then take the square root: $a = \sqrt{c^2 - b^2}$.

The Converse of Pythagoras: This concept is used to verify if a triangle is right-angled. If the side lengths of a triangle satisfy the equation $a^2 + b^2 = c^2$, then the triangle must have a $90^{\circ}$ angle opposite the longest side. If the equation does not balance, the triangle is not right-angled.

Pythagorean Triples: These are sets of three positive integers $(a, b, c)$ that perfectly satisfy the theorem. Common examples include $(3, 4, 5)$, $(5, 12, 13)$, and $(8, 15, 17)$. Multiples of these sets (e.g., $6, 8, 10$) are also triples.

3D Applications: The theorem can be applied to find the space diagonal of a cuboid. Visually, this is a line connecting one bottom corner to the opposite top corner. It involves two steps: first finding the diagonal of the base using $d_{base} = \sqrt{l^2 + w^2}$, then using that to find the space diagonal: $D = \sqrt{d_{base}^2 + h^2}$, which simplifies to $D = \sqrt{l^2 + w^2 + h^2}$.