Geometry - Pythagoras Theorem and its Converse

Right-Angled Triangle: A triangle in which one angle is exactly $90^\circ$. Visually, this is often indicated by a small square symbol at the vertex where the two perpendicular sides (the base and the height) meet. The side opposite to this $90^\circ$ angle is the longest side, known as the hypotenuse.

The Pythagoras Theorem: In any right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. If the sides are $a$ and $b$ and the hypotenuse is $c$, then $c^2 = a^2 + b^2$. Visually, if you draw a square on each side of the triangle, the area of the largest square is equal to the combined area of the two smaller squares.

Converse of Pythagoras Theorem: If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle must be a right-angled triangle. The angle opposite the longest side will be the $90^\circ$ angle.

Pythagorean Triplets: A set of three positive integers $(a, b, c)$ that satisfy the condition $a^2 + b^2 = c^2$. Common examples include $(3, 4, 5)$, $(5, 12, 13)$, and $(8, 15, 17)$. Any multiple of a Pythagorean triplet, such as $(6, 8, 10)$, is also a Pythagorean triplet.

Diagonal of a Rectangle and Square: In a rectangle with length $l$ and breadth $b$, the diagonal $d$ creates two right-angled triangles, where $d = \sqrt{l^2 + b^2}$. In a square with side $s$, the diagonal $d$ forms a right-angled triangle where the legs are equal, resulting in $d = s\sqrt{2}$.

Identifying Triangle Types: Using the side lengths $a, b,$ and $c$ (where $c$ is the longest side), we can classify triangles: if $c^2 = a^2 + b^2$, it is right-angled; if $c^2 > a^2 + b^2$, the triangle is obtuse-angled (one angle $> 90^\circ$); if $c^2 < a^2 + b^2$, the triangle is acute-angled (all angles $< 90^\circ$).