Right Triangle
A right triangle has one angle measuring exactly $90^\circ$, and its sides satisfy the Pythagorean theorem: $a^2 + b^2 = c^2$.
Formula
a^2 + b^2 = c^2 \text{ (Pythagorean theorem)}
Definition
A right triangle has one corner that is exactly a right angle ($90^\circ$); the two sides forming it are the legs, and the side opposite the right angle is the hypotenuse. The Pythagorean theorem states $a^2+b^2=c^2$, the two acute angles are complementary, and trigonometric ratios (sin, cos, tan) are defined using right triangles. It can be inscribed in a semicircle (Thales' theorem: any triangle inscribed in a semicircle is a right triangle), and geometric mean relationships hold: the altitude to the hypotenuse $h$ satisfies $h^2=pq$, with $a^2=pc$ and $b^2=qc$ for the two hypotenuse segments $p, q$.
Example
A triangle with sides $3, 4$, and $5$ is a right triangle: $3^2+4^2=9+16=25=5^2$, with $5$ as the hypotenuse; a ramp against a wall forms a right triangle. A $45$-$45$-$90$ triangle with legs $1$ has hypotenuse $\sqrt{2}$, and a $30$-$60$-$90$ triangle with short leg $1$ has long leg $\sqrt{3}$ and hypotenuse $2$. For legs $6, 8$ and hypotenuse $10$: the altitude to the hypotenuse is $h=(6\cdot8)/10=4.8$, with hypotenuse segments $p=3.6$ and $q=6.4$, and indeed $pq=23.04=4.8^2$.
Key Insight
The Pythagorean theorem only works for right triangles, and this special relationship between the sides is what makes right triangles the foundation of all of trigonometry, since every trig ratio is defined as a ratio of sides in one. Thales' theorem, that any angle inscribed in a semicircle is a right angle, is the converse of the inscribed angle theorem for $180^\circ$ arcs, showing that the set of right triangles inscribed on a given diameter is exactly the semicircle itself, connecting the algebraic Pythagorean condition to the geometric circle.