Geometry and Measurement - The Pythagorean Theorem

A Right-Angled Triangle is a triangle with one interior angle measuring exactly $90^{\circ}$. Visually, this is represented by a small square symbol in the corner where the two perpendicular sides (the legs) meet.

The Hypotenuse is the longest side of a right-angled triangle. It is always located directly opposite the right angle ($90^{\circ}$). In a diagram, if you draw an arrow pointing out from the right-angle corner, it will always point toward the hypotenuse.

The Legs are the two shorter sides of the triangle that form the right angle. In the standard formula, these are labeled as $a$ and $b$. Visually, these sides are perpendicular to each other.

The Pythagorean Theorem states that the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. Geometrically, this means if you drew a square on each side of the triangle, the combined area of the two smaller squares would perfectly equal the area of the largest square.

Calculating the Hypotenuse involves squaring both legs, adding them together, and then taking the square root. Since $c^2 = a^2 + b^2$, we find $c$ by calculating $\sqrt{a^2 + b^2}$.

Finding a Missing Leg requires subtracting the square of the known leg from the square of the hypotenuse. If the hypotenuse $c$ and leg $b$ are known, the missing leg $a$ is found using $a = \sqrt{c^2 - b^2}$.

Pythagorean Triples are sets of three whole numbers that satisfy the theorem perfectly, such as $(3, 4, 5)$ or $(5, 12, 13)$. If a triangle's side lengths match a triple or a multiple of a triple (like $6, 8, 10$), it is guaranteed to be a right-angled triangle.

The Converse of the Pythagorean Theorem is used to verify if a triangle is right-angled. If you measure all three sides and find that $a^2 + b^2 = c^2$ is true, the triangle contains a $90^{\circ}$ angle; if the equation is not equal, the triangle is not right-angled.